{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First composition derivatives from SRK\n",
    "======================================"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{} \\alpha^r}{{}{\\partial \\delta}} = \\frac{1}{R T_{r}} \\left(- R T_{r}\\frac{b}{b\\rho_r\\delta - 1} - \\rho_{r} \\tau\\frac{ a{\\left (\\tau \\right )}}{b \\delta \\rho_{r} + 1}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{} \\alpha^r}{{}{\\partial \\delta}}\\right) = \\frac{1}{R T_{r}} \\left(-RT_r \\frac{(b\\delta\\rho_r-1)\\frac{db}{dx_i}-b\\rho_r\\delta\\frac{db}{dx_i}}{(b \\rho_{r}\\delta - 1)^2} - \\rho_{r} \\tau \\frac{(b \\delta \\rho_{r} + 1)\\frac{\\partial a}{\\partial x_i}-a\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{(b \\delta \\rho_{r} + 1)^2}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{} \\alpha^r}{{}{\\partial \\delta}}\\right) = \\frac{1}{R T_{r}} \\left( \\frac{db}{dx_i}\\frac{RT_r}{(b \\rho_{r}\\delta - 1)^2} - \\rho_{r} \\tau \\frac{(b \\delta \\rho_{r} + 1)\\frac{\\partial a}{\\partial x_i}-a\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{(b \\delta \\rho_{r} + 1)^2}\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{} \\alpha^r}{{\\partial \\tau}{}} = \\frac{1}{R T_{r}} \\left(\\left(- \\tau\\frac{\\frac{d}{d \\tau} a{\\left (\\tau \\right )}}{b} - \\frac{a{\\left (\\tau \\right )}}{b} \\right) \\log{\\left (b \\delta \\rho_{r} + 1 \\right )}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{} \\alpha^r}{{\\partial \\tau}{}}\\right) = \\frac{1}{R T_{r}} \\left(\\left(- \\tau \\frac{b\\frac{d^2a}{\\partial\\tau\\partial x_i} -\\frac{\\partial a}{\\partial \\tau}\\frac{\\partial b}{\\partial x_i}}{b^2} - \\frac{b\\frac{\\partial a}{\\partial x_i}-a\\frac{\\partial b}{\\partial x_i}}{b^2} \\right) \\log{\\left (b \\delta \\rho_{r} + 1 \\right )} + \\left(- \\tau\\frac{\\frac{d}{d \\tau} a{\\left (\\tau \\right )}}{b} - \\frac{a{\\left (\\tau \\right )}}{b} \\right) \\frac{\\delta\\rho_r}{\\left (b \\delta \\rho_{r} + 1 \\right )}\\frac{\\partial b}{\\partial x_i} \\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{2}} \\alpha^r}{{}{\\partial \\delta^{2}}} = \\frac{1}{R T_{r}} \\left(R T_{r}\\frac{1}{\\left(\\delta - \\frac{1}{b \\rho_{r}}\\right)^{2}} + \\rho_{r}^{2} \\tau \\frac{b a{\\left (\\tau \\right )}}{\\left(b \\delta \\rho_{r} + 1\\right)^{2}}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i} \\left( \\frac{\\partial{^{2}} \\alpha^r}{{}{\\partial \\delta^{2}}} \\right) = \\frac{1}{R T_{r}} \\left(R T_{r}\\frac{-2\\left(\\delta - \\frac{1}{b \\rho_{r}}\\right)\\left(\\frac{1}{b\\rho_r^2}\\right)}{\\left(\\delta - \\frac{1}{b \\rho_{r}}\\right)^{4}} + \\rho_{r}^{2} \\tau \\frac{\\left(b \\delta \\rho_{r} + 1\\right)^{2}(b\\frac{\\partial a}{\\partial x_i}-a\\frac{\\partial b}{\\partial x_i})-2ba\\left(b \\delta \\rho_{r} + 1\\right)\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{\\left(b \\delta \\rho_{r} + 1\\right)^{2}}\\right) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i} \\left( \\frac{\\partial{^{2}} \\alpha^r}{{}{\\partial \\delta^{2}}} \\right) = \\frac{1}{R T_{r}} \\left(-\\frac{2R T_{r}}{b^2\\rho_r\\left(\\delta - \\frac{1}{b \\rho_{r}}\\right)^{3}} + \\rho_{r}^{2} \\tau\\left( \\left(b\\frac{\\partial a}{\\partial x_i}-a\\frac{\\partial b}{\\partial x_i}\\right) -\\frac{2ba\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{\\left(b \\delta \\rho_{r} + 1\\right)}\\right)\\right) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{2}} \\alpha^r}{{\\partial \\tau}{\\partial \\delta}} = -\\frac{\\rho_r}{R T_{r} } \\left(\\frac{\\tau \\frac{da}{d \\tau}+a}{b \\delta \\rho_{r} + 1} \\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{2}} \\alpha^r}{{\\partial \\tau}{\\partial \\delta}}\\right) = -\\frac{\\rho_r}{R T_{r} } \\left( \\frac{(b \\delta \\rho_{r} + 1)(\\tau \\frac{\\partial^2a}{\\partial \\tau \\partial x_i}+\\frac{\\partial a}{\\partial x_i})-(\\tau \\frac{da}{d \\tau}+a)\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{(b \\delta \\rho_{r} + 1)^2} \\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{2}} \\alpha^r}{{\\partial \\tau^{2}}{}} = - \\frac{1}{R T_{r} } \\left(\\tau \\frac{d^{2}}{d \\tau^{2}}  a{\\left (\\tau \\right )} + 2 \\frac{d}{d \\tau} a{\\left (\\tau \\right )}\\right) \\frac{\\log{\\left (b \\delta \\rho_{r} + 1 \\right )}}{b}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{2}} \\alpha^r}{{\\partial \\tau^{2}}{}}\\right) = - \\frac{1}{R T_{r} } \\left[\\left(\\tau \\frac{\\partial^{3}a}{\\partial \\tau^{2}\\partial x_i}   + 2 \\frac{\\partial^2a}{\\partial \\tau\\partial x_i} \\right) \\frac{\\log{\\left (b \\delta \\rho_{r} + 1 \\right )}}{b} + \\left(\\tau \\frac{d^{2}}{d \\tau^{2}}  a{\\left (\\tau \\right )} + 2 \\frac{d}{d \\tau} a{\\left (\\tau \\right )}\\right) \\frac{\\partial b}{\\partial x_i}\\frac{\\frac{b\\delta\\rho_r}{\\left (b \\delta \\rho_{r} + 1 \\right )} - \\log{\\left (b \\delta \\rho_{r} + 1 \\right )}}{b^2} \\right]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{3}} \\alpha^r}{{}{\\partial \\delta^{3}}} = - \\frac{1}{R T_{r}} \\left(2 R T_{r}\\frac{1}{\\left(\\delta - \\frac{1}{b \\rho_{r}}\\right)^{3}} + 2 \\rho_{r}^{3} \\tau \\frac{b^{2} a{\\left (\\tau \\right )}}{\\left(b \\delta \\rho_{r} + 1\\right)^{3}}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{3}} \\alpha^r}{{}{\\partial \\delta^{3}}}\\right) = - \\frac{1}{R T_{r}} \\left(2 R T_{r}\\frac{-3\\frac{1}{b^2\\rho_r}\\frac{\\partial b}{\\partial x_i}}{\\left(\\delta - \\frac{1}{b \\rho_{r}}\\right)^{4}} + 2 \\rho_{r}^{3} \\tau \\frac{\\left(b \\delta \\rho_{r} + 1\\right)^{3}(2ba\\frac{\\partial b}{\\partial x_i}+b^2\\frac{\\partial a}{\\partial x_i})-(b^2a)3\\left(b \\delta \\rho_{r} + 1\\right)^{2}\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{\\left(b \\delta \\rho_{r} + 1\\right)^{6}}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{3}} \\alpha^r}{{}{\\partial \\delta^{3}}}\\right) = - \\frac{1}{R T_{r}} \\left(\\frac{-6 R T_{r}}{b^2\\rho_r}\\frac{\\frac{\\partial b}{\\partial x_i}}{\\left(\\delta - \\frac{1}{b \\rho_{r}}\\right)^{4}} + 2 \\rho_{r}^{3} \\tau \\frac{\\left(b \\delta \\rho_{r} + 1\\right)(2ba\\frac{\\partial b}{\\partial x_i}+b^2\\frac{\\partial a}{\\partial x_i})-3b^2a\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{\\left(b \\delta \\rho_{r} + 1\\right)^{4}}\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{3}} \\alpha^r}{{\\partial \\tau}{\\partial \\delta^{2}}} = \\frac{\\rho_{r}^{2}}{R T_{r} }\\frac{b  \\left(\\tau \\frac{d}{d \\tau} a{\\left (\\tau \\right )} + a{\\left (\\tau \\right )}\\right)}{\\left(b \\delta \\rho_{r} + 1\\right)^{2}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{3}} \\alpha^r}{{\\partial \\tau}{\\partial \\delta^{2}}}\\right) = \\frac{\\rho_{r}^{2}}{R T_{r} }  \\frac{\\left(b \\delta \\rho_{r} + 1\\right)^2\\left[\\frac{\\partial b}{\\partial x_i}\\left(\\tau \\frac{da}{d \\tau}  + a\\right) + b\\left(\\tau \\frac{d^2a}{\\partial \\tau\\partial x_i}  + \\frac{\\partial a}{\\partial x_i}\\right)\\right] -2b\\delta \\rho_{r} \\left(b \\delta \\rho_{r} + 1\\right) \\left(\\tau \\frac{da}{d \\tau}  + a\\right) \\frac{\\partial b}{\\partial x_i}}{\\left(b \\delta \\rho_{r} + 1\\right)^{4}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{3}} \\alpha^r}{{\\partial \\tau^{2}}{\\partial \\delta}} = - \\frac{\\rho_{r}}{R T_{r}}\\frac{1}{ \\left(b \\delta \\rho_{r} + 1\\right)} \\left(\\tau \\frac{d^{2}}{d \\tau^{2}}  a{\\left (\\tau \\right )} + 2 \\frac{d}{d \\tau} a{\\left (\\tau \\right )}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{3}} \\alpha^r}{{\\partial \\tau^{2}}{\\partial \\delta}}\\right) = - \\frac{\\rho_{r}}{R T_{r}}\\left[\\frac{1}{ \\left(b \\delta \\rho_{r} + 1\\right)} \\left(\\tau \\frac{d^{3}a}{d \\tau^{2}\\partial x_i} + 2 \\frac{\\partial^2a}{\\partial\\tau\\partial x_i} \\right) + \\frac{-\\delta \\rho_{r}}{ \\left(b \\delta \\rho_{r} + 1\\right)^2}\\frac{\\partial b}{\\partial x_i} \\left(\\tau \\frac{d^{2}}{d \\tau^{2}}  a{\\left (\\tau \\right )} + 2 \\frac{d}{d \\tau} a{\\left (\\tau \\right )}\\right)\\right]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{3}} \\alpha^r}{{\\partial \\tau^{3}}{}} = - \\frac{1}{R T_{r}} \\left(\\tau \\frac{d^{3}}{d \\tau^{3}}  a{\\left (\\tau \\right )} + 3 \\frac{d^{2}}{d \\tau^{2}}  a{\\left (\\tau \\right )}\\right) \\frac{\\log{\\left (b \\delta \\rho_{r} + 1 \\right )}}{b} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{3}} \\alpha^r}{{\\partial \\tau^{3}}{}}\\right) = - \\frac{1}{R T_{r} } \\left[\\left(\\tau \\frac{\\partial^{4}a}{\\partial \\tau^{3}\\partial x_i}   + 3 \\frac{\\partial^3a}{\\partial \\tau^2\\partial x_i} \\right) \\frac{\\log{\\left (b \\delta \\rho_{r} + 1 \\right )}}{b} + \\left(\\tau \\frac{d^{3}}{d \\tau^{3}}  a{\\left (\\tau \\right )} + 3 \\frac{d^2}{d \\tau^2} a{\\left (\\tau \\right )}\\right) \\frac{\\partial b}{\\partial x_i}\\frac{\\frac{b\\delta\\rho_r}{\\left (b \\delta \\rho_{r} + 1 \\right )} - \\log{\\left (b \\delta \\rho_{r} + 1 \\right )}}{b^2} \\right]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{4}} \\alpha^r}{{}{\\partial \\delta^{4}}} = - \\frac{1}{R T_{r}} \\left(6 R T_{r}\\frac{1}{\\left(\\delta - \\frac{1}{b \\rho_{r}}\\right)^{4}} + 6 \\rho_{r}^{4} \\tau \\frac{b^{3} a{\\left (\\tau \\right )}}{\\left(b \\delta \\rho_{r} + 1\\right)^{4}}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{4}} \\alpha^r}{{}{\\partial \\delta^{4}}}\\right) = - \\frac{1}{R T_{r}} \\left(6 R T_{r}\\frac{-4\\frac{1}{b^2\\rho_r}\\frac{\\partial b}{\\partial x_i}}{\\left(\\delta - \\frac{1}{b \\rho_{r}}\\right)^{5}} + 6 \\rho_{r}^{4} \\tau \\frac{\\left(b \\delta \\rho_{r} + 1\\right)^{4}(3b^2a\\frac{\\partial b}{\\partial x_i}+b^3\\frac{\\partial a}{\\partial x_i})-(b^3a)4\\left(b \\delta \\rho_{r} + 1\\right)^{3}\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{\\left(b \\delta \\rho_{r} + 1\\right)^{8}}\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{4}} \\alpha^r}{{\\partial \\tau}{\\partial \\delta^{3}}} = - \\frac{2 \\rho_{r}^{3}}{R T_{r} }\\frac{ b^{2}}{\\left(b \\delta \\rho_{r} + 1\\right)^{3}} \\left(\\tau \\frac{d}{d \\tau} a{\\left (\\tau \\right )} + a{\\left (\\tau \\right )}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{4}} \\alpha^r}{{\\partial \\tau}{\\partial \\delta^{3}}}\\right) = - \\frac{2 \\rho_{r}^{3}}{R T_{r} }\\left[\\frac{b^{2}}{\\left(b \\delta \\rho_{r} + 1\\right)^{3}} \\left(\\tau \\frac{\\partial^2a}{\\partial\\tau\\partial x_i}  + \\frac{\\partial a}{\\partial x_i}\\right) + \\frac{\\left(b \\delta \\rho_{r} + 1\\right)^{3}2b\\frac{db}{dx_i}-b^23\\left(b \\delta \\rho_{r} + 1\\right)^2\\delta\\rho_r\\frac{db}{dx_i}}{\\left(b \\delta \\rho_{r} + 1\\right)^{6}} \\left(\\tau \\frac{d}{d \\tau} a{\\left (\\tau \\right )} + a{\\left (\\tau \\right )}\\right)\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{4}} \\alpha^r}{{\\partial \\tau}{\\partial \\delta^{3}}}\\right) = - \\frac{2 \\rho_{r}^{3}}{R T_{r} }\\left[\\frac{b^{2}}{\\left(b \\delta \\rho_{r} + 1\\right)^{3}} \\left(\\tau \\frac{\\partial^2a}{\\partial\\tau\\partial x_i}  + \\frac{\\partial a}{\\partial x_i}\\right) + b\\frac{\\partial b}{\\partial x_i}\\frac{2\\left(\\delta \\rho_{r} + 1\\right)-3b\\delta\\rho_r}{\\left(b \\delta \\rho_{r} + 1\\right)^{4}} \\left(\\tau \\frac{d}{d \\tau} a{\\left (\\tau \\right )} + a{\\left (\\tau \\right )}\\right)\\right]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{4}} \\alpha^r}{{\\partial \\tau^{2}}{\\partial \\delta^{2}}} = \\frac{\\rho_{r}^{2}}{R T_{r}}\\frac{b}{ \\left(b \\delta \\rho_{r} + 1\\right)^{2}} \\left(\\tau \\frac{d^{2}}{d \\tau^{2}}  a{\\left (\\tau \\right )} + 2 \\frac{d}{d \\tau} a{\\left (\\tau \\right )}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{4}} \\alpha^r}{{\\partial \\tau^{2}}{\\partial \\delta^{2}}}\\right) = \\frac{\\rho_{r}^{2}}{R T_{r}}\\left[\\frac{b}{\\left(b \\delta \\rho_{r} + 1\\right)^{2}} \\left(\\tau \\frac{\\partial^{3}a}{\\partial \\tau^{2}\\partial x_i} + 2 \\frac{d^2a}{d \\tau\\partial x_i}\\right) + \\frac{\\left(b \\delta \\rho_{r} + 1\\right)^{2}\\frac{\\partial b}{\\partial x_i} -b2\\left(b \\delta \\rho_{r} + 1\\right)\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{ \\left(b \\delta \\rho_{r} + 1\\right)^{4}} \\left(\\tau \\frac{d^{2}}{d \\tau^{2}}  a{\\left (\\tau \\right )} + 2 \\frac{d}{d \\tau} a{\\left (\\tau \\right )}\\right)\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{4}} \\alpha^r}{{\\partial \\tau^{2}}{\\partial \\delta^{2}}}\\right) = \\frac{\\rho_{r}^{2}}{R T_{r}}\\left[\\frac{b}{\\left(b \\delta \\rho_{r} + 1\\right)^{2}} \\left(\\tau \\frac{\\partial^{3}a}{\\partial \\tau^{2}\\partial x_i} + 2 \\frac{d^2a}{d \\tau\\partial x_i}\\right) + \\frac{\\left(b \\delta \\rho_{r} + 1\\right)\\frac{\\partial b}{\\partial x_i} -2b\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{ \\left(b \\delta \\rho_{r} + 1\\right)} \\left(\\tau \\frac{d^{2}}{d \\tau^{2}}  a{\\left (\\tau \\right )} + 2 \\frac{d}{d \\tau} a{\\left (\\tau \\right )}\\right)\\right]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{4}} \\alpha^r}{{\\partial \\tau^{3}}{\\partial \\delta}} = - \\frac{\\rho_{r}}{R T_{r}}\\frac{\\tau \\frac{d^{3}a}{d \\tau^{3}} + 3 \\frac{d^{2}a}{d \\tau^{2}}}{ \\left(b \\delta \\rho_{r} + 1\\right)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{4}} \\alpha^r}{{\\partial \\tau^{3}}{\\partial \\delta}}\\right) = - \\frac{\\rho_{r}}{R T_{r}}\\frac{\\left(b \\delta \\rho_{r} + 1\\right)\\left(\\tau \\frac{d^{4}a}{d \\tau^{3}\\partial x_i} + 3 \\frac{\\partial^{3}a}{\\partial \\tau^{2}\\partial x_i}\\right) - \\left(\\tau \\frac{d^{3}a}{d \\tau^{3}} + 3 \\frac{d^{2}a}{d \\tau^{2}}\\right)\\delta\\rho_r\\frac{\\partial b}{\\partial x_i}}{ \\left(b \\delta \\rho_{r} + 1\\right)^2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial{^{4}} \\alpha^r}{{\\partial \\tau^{4}}{}} = - \\frac{1}{R T_{r}} \\left(\\tau \\frac{d^{4}}{d \\tau^{4}}  a{\\left (\\tau \\right )} + 4 \\frac{d^{3}}{d \\tau^{3}}  a{\\left (\\tau \\right )}\\right) \\frac{\\log{\\left (b \\delta \\rho_{r} + 1 \\right )}}{b}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial}{\\partial x_i}\\left(\\frac{\\partial{^{4}} \\alpha^r}{{\\partial \\tau^{4}}{}}\\right) = - \\frac{1}{R T_{r} } \\left[\\left(\\tau \\frac{\\partial^{5}a}{\\partial \\tau^{4}\\partial x_i}   + 4 \\frac{\\partial^4a}{\\partial \\tau^3\\partial x_i} \\right) \\frac{\\log{\\left (b \\delta \\rho_{r} + 1 \\right )}}{b} + \\left(\\tau \\frac{d^{4}}{d \\tau^{4}}  a{\\left (\\tau \\right )} + 4 \\frac{d^3}{d \\tau^3} a{\\left (\\tau \\right )}\\right) \\frac{\\partial b}{\\partial x_i}\\frac{\\frac{b\\delta\\rho_r}{\\left (b \\delta \\rho_{r} + 1 \\right )} - \\log{\\left (b \\delta \\rho_{r} + 1 \\right )}}{b^2} \\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Derivatives of $a$\n",
    "=================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$a = \\sum_i\\sum_jx_ix_j\\sqrt{a_ia_j}(1-k_{ij}) $$\n",
    "\n",
    "$$\\frac{\\partial a}{\\partial \\tau} = \\sum_i\\sum_jx_ix_j\\frac{1}{2\\sqrt{a_ia_j}}\\left(a_i\\frac{\\partial a_j}{\\partial \\tau}+a_j\\frac{\\partial a_i}{\\partial \\tau}\\right)(1-k_{ij}) $$\n",
    "\n",
    "$$\\frac{\\partial^2 a}{\\partial \\tau\\partial x_i} = \\sum_jx_j\\frac{1}{2\\sqrt{a_ia_j}}\\left(a_i\\frac{\\partial a_j}{\\partial \\tau}+a_j\\frac{\\partial a_i}{\\partial \\tau}\\right)(1-k_{ij}) $$\n",
    "\n",
    "$$\\frac{\\partial^3 a}{\\partial \\tau\\partial x_i\\partial x_j} = \\frac{1}{2\\sqrt{a_ia_j}}\\left(a_i\\frac{\\partial a_j}{\\partial \\tau}+a_j\\frac{\\partial a_i}{\\partial \\tau}\\right)(1-k_{ij}) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHYAAAAmCAMAAAAm95/UAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\nH7fcZDo0X/ZZ1P3Cv+G77BfYAP4BfjoWYGaPaKwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sqrt{\\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )}}$$"
      ],
      "text/plain": [
       "  ______________\n",
       "╲╱ aᵢ(τ)⋅a_j(τ) "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAasAAAA2CAMAAABOQEZ2AAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{\\sqrt{\\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )}}}{2 \\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )}} \\left(\\operatorname{a_{i}}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )} + \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ______________ ⎛      d                   d        ⎞\n",
       "╲╱ aᵢ(τ)⋅a_j(τ) ⋅⎜aᵢ(τ)⋅──(a_j(τ)) + a_j(τ)⋅──(aᵢ(τ))⎟\n",
       "                 ⎝      dτ                  dτ       ⎠\n",
       "──────────────────────────────────────────────────────\n",
       "                    2⋅aᵢ(τ)⋅a_j(τ)                    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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IxMBRTbssrqF1jrgIvH9DAKOxYIdMaU5Zb5e0SHKBkqanS8mSZQKDHCwepjiiTLjjFXsH\niI+WR42EWkukJkQupzeZZ/AJEYVkJ3muKUQairncGmrpTVuRJW0HFa4Qt5gpKNk+6ODlf8JMcRBX\nN8gSCxFNEaa+MBUlc9kMAYuYlNEY1Hl5wMJCAjXFVOEnRCQ/VN2XEgYkvCXREneMsM2TJie0cKbY\njmraRQ4Ibh98w/QHLEwjZKBIc1P6WTTi781xWS+nxynjDZFyBxtZqo/+rJZHjYY6R8TftUwJgXaW\nZiMHl0iyYi63hpILJJ7wbWeqSFtVLGYKzGwp30BNgS5uFJCe9MuEWVLdIE0sRDxFmPqiq+gzV6NG\nTNKy6nmweUwTqNyxPlOFnyfi80P/2B89wiMFzaMVR0vcMZ5tqFuWtHAq9jHEUU27yAHBpg0jfK9u\naECyccckNyXPaNFVUE45QfM/LuvlIDkkImUQWfo01ORlBlLzqNFQ+4nUPJHuN7tRhy0RQvQCjXym\nKVR8KOdyM6slibT+5afYTmRYd+57gitoJu//51HzHCTUDXZSn6LpnjDzha2nMNeixp2dhkUTYqUE\napqpZ4gnovAjFiUMyPEWQStrJ0xQHqjhTLAd07TLHBBq2q37qb/j80oWW0tSdCuikkWDJ7kQwglZ\nL6fHK/KIIqDFofe+77vdStAIpUUe3goWSohaUIZLJTPKnqGiN1PHlH3buaaoxUJB/qQh1xE4SvEZ\nxKiFiOQJe4H2mYeihqh+jlkPllTOLnrGCscOQJRpWsTnF7bHQ/NopdDKbAtbwm49qObZJxw0qmkD\nuil4CoQDQk3748YiH02kc6RPvjiASvTzJBfkgpkqIEN2qqzM6XETP2vD0OIHIiuakkfNLheLJVLz\nRa7E/k47JYTmFOOiaF5JcGuEpfjDuQTOs51cS3lCKihYi0u1BMsJPlZO+hSxhLDwhannMdejhohd\nKQkr6ucdClTpWAsimSZFPH4xUwRahrcoWrFtVJiWJRBcE/ZJB5U3bUAeCkKBdECoad+GbD2mPnaO\nMrkHy03pvoEDSTTKZWVOj08ogZCkPOq4JJHa8fx4fbqXfEKWOQNNKMFFqXqll1gFrtrO1BVaHB11\nYcDVDgotHPSqzANRq2bqRKASpiq/0frTaCW2Rc0YC1TetKNm+BdDTXt4h23NYDt0jphFo89twXNT\nDos2A0k0ULZP5ZEh61bEOqNb+bM27sLS/y92AOFkOx1ClhNa2qSwvr4Th+uq7XB1RGHVn2YtsVdl\nvtWolRAb6qr8RuD0InXRRpsRFVytaR/6qWH7w6PYOb7N9rtuxLfPbcFyUw7pi0JJNGQGjgxZL9/I\nnUy1Rb0278X2bedgD2bag5LlhOa1oAid/fK5bnsRnqis/T6mqLKJQ535ZqNW7DOdXzHMIFAXbawV\nKbm1mva164hudjMydI5qAhsggBvF9SQaSmKQuKy3pX7osEFqpUK/Xfk+/GA1IbuSPQm1B/qVUt/2\nkzcvlbBnpcv1ma9EJKC2Lr+6aAGTJ59eq2l3k9Vt1wFA54hZNEhuC2CInaOWRANlQYAUVFmvc7wV\n/XA1ga9b7EYiWre4mpCtq6YW2sVZagHr2z56CUEtfpk49ZlnKl6oWl1+ddHmcsFKTfvT/WrS7Ww/\n3/ew7w7ScdDcFkAcs2hkJ7+Jy9qpOvY5zjojw1RFDtx+ebOq0tYiZCNCa16647R6fdufYr5nTaIx\n3fWZx7Qtf60uv7po83ljnab9Iguuu28RQ9B5rF7yG/AaZOCg+VTkhIx57hG9JcgvWWg715yuzc3u\nSmQGL2lGvi7Sf9W3/YpZNfMtWqFmfeYrkIiorMuvLlrE7ImX1mnadIWPW/FIEtjQTTiOH2TRUJNo\ndD2sTH7jRPXEOX6+kdCiTMBZpHA1M/jP+7npxhwo2UW0j1BCNoHXtv3i/UblCPsWEanNfBGjC5TU\n5VcXrYBGYdVVmjYuPn+cvt9zv0UW03FonSO8BOcnvwFPqLJ+vpHj+GXgoGp6wSzLenyO51f36wuU\n7HToeRDI5uHatp/wlX0e26uh1mZezbBKQHX51UWrRFGBWaNpB/Z0g3Va55iVvMQi0D1HDjGVgWOo\n1416Opn9f64HirZN5ILaerd3dp7xEti97u6BTA+s0LRTP+yldo7KRiwvy4elrHWOiizLwDG46pmR\nmCfTq/9SteNMbmtxlcG/5M6d63Y8sELT7uaqwx7guS2gnti2baZwaH4YqKZ1jmJLua2rCm/ivRqY\n/EyB/KBlTZt/56W6Jusda0seWLxpj3xASyW5GHyqdo6erNJdGvnHj6wc2VLzsbYc5phUfkbSNW3N\nAbs9f9UDSzftq/rEl/ZuIsmFBRibgcMpfw8bU9zx/j/PAzNMHF1GtpI8g/dauwfyPLBs077MNERl\nuZppsEkTnGRlU57r9lq9Bw7d5HpFb7TnfTKmojt3qNEeWLRpzzYDNJo+EXR7msmpvbh7YPfA7oEl\nPEDWDi2hrlDH52eWHRcS26vvHtg9sHUPdBm5tmtkn0tyu/btlu0e2D3wRz1wnmNis6avzpMGLWta\nsmPtHtg98A95YKZ1QzU9GPlhmppqdqzdA7sHdg+gB47Kbwji1W2UtGyS27Bst2L3wO6BP+uBo83D\nNXz+B5iO8YyzdsW0AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{\\sqrt{\\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )}}}{4 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )}} \\left(2 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} - \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )}\\right)^{2} + 2 \\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} + 2 \\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )} - \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )}\\right)^{2}\\right)$$"
      ],
      "text/plain": [
       "                 ⎛                  2                              2          \n",
       "  ______________ ⎜    2            d              2    ⎛d         ⎞           \n",
       "╲╱ aᵢ(τ)⋅a_j(τ) ⋅⎜2⋅aᵢ (τ)⋅a_j(τ)⋅───(a_j(τ)) - aᵢ (τ)⋅⎜──(a_j(τ))⎟  + 2⋅aᵢ(τ)\n",
       "                 ⎜                  2                  ⎝dτ        ⎠           \n",
       "                 ⎝                dτ                                          \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                            2 \n",
       "                                                                        4⋅aᵢ (\n",
       "\n",
       "           2                                                                  \n",
       "    2     d                          d         d               2    ⎛d        \n",
       "⋅a_j (τ)⋅───(aᵢ(τ)) + 2⋅aᵢ(τ)⋅a_j(τ)⋅──(aᵢ(τ))⋅──(a_j(τ)) - a_j (τ)⋅⎜──(aᵢ(τ))\n",
       "           2                         dτ        dτ                   ⎝dτ       \n",
       "         dτ                                                                   \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "      2                                                                       \n",
       "τ)⋅a_j (τ)                                                                    \n",
       "\n",
       " 2⎞\n",
       "⎞ ⎟\n",
       "⎟ ⎟\n",
       "⎠ ⎟\n",
       "  ⎠\n",
       "───\n",
       "   \n",
       "   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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1O/rX6mHyCcVZImlLdhmajH4Lli5jlJCabD/yPkdpDxE7B/E/gm3Hjc8/T3et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\nNAWaAjtR4Eim8/8DIzoF1xyZHA8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\sqrt{\\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )}}}{8 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )}} \\left(4 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\frac{d^{3}}{d \\tau^{3}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} - 6 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} + 3 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )}\\right)^{3} + 4 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )} \\frac{d^{3}}{d \\tau^{3}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} + 6 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} + 6 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} - 3 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )}\\right)^{2} - 6 \\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} - 3 \\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )}\\right)^{2} \\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )} + 3 \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )}\\right)^{3}\\right)$$"
      ],
      "text/plain": [
       "                 ⎛                   3                                        \n",
       "  ______________ ⎜    3       2     d                3           d           d\n",
       "╲╱ aᵢ(τ)⋅a_j(τ) ⋅⎜4⋅aᵢ (τ)⋅a_j (τ)⋅───(a_j(τ)) - 6⋅aᵢ (τ)⋅a_j(τ)⋅──(a_j(τ))⋅──\n",
       "                 ⎜                   3                           dτ           \n",
       "                 ⎝                 dτ                                       dτ\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "2                                3                      3                     \n",
       "                3    ⎛d         ⎞        2       3     d               2      \n",
       "─(a_j(τ)) + 3⋅aᵢ (τ)⋅⎜──(a_j(τ))⎟  + 4⋅aᵢ (τ)⋅a_j (τ)⋅───(aᵢ(τ)) + 6⋅aᵢ (τ)⋅a_\n",
       "2                    ⎝dτ        ⎠                       3                     \n",
       "                                                      dτ                      \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                  2                                         2                 \n",
       " 2    d          d                2       2    d           d               2  \n",
       "j (τ)⋅──(aᵢ(τ))⋅───(a_j(τ)) + 6⋅aᵢ (τ)⋅a_j (τ)⋅──(a_j(τ))⋅───(aᵢ(τ)) - 3⋅aᵢ (τ\n",
       "      dτ          2                            dτ           2                 \n",
       "                dτ                                        dτ                  \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                           3       3                                          \n",
       "                       8⋅aᵢ (τ)⋅a_j (τ)                                       \n",
       "\n",
       "                               2                               2              \n",
       "         d         ⎛d         ⎞               3    d          d               \n",
       ")⋅a_j(τ)⋅──(aᵢ(τ))⋅⎜──(a_j(τ))⎟  - 6⋅aᵢ(τ)⋅a_j (τ)⋅──(aᵢ(τ))⋅───(aᵢ(τ)) - 3⋅aᵢ\n",
       "         dτ        ⎝dτ        ⎠                    dτ          2              \n",
       "                                                             dτ               \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                       2                                   3⎞\n",
       "       2    ⎛d        ⎞  d                 3    ⎛d        ⎞ ⎟\n",
       "(τ)⋅a_j (τ)⋅⎜──(aᵢ(τ))⎟ ⋅──(a_j(τ)) + 3⋅a_j (τ)⋅⎜──(aᵢ(τ))⎟ ⎟\n",
       "            ⎝dτ       ⎠  dτ                     ⎝dτ       ⎠ ⎟\n",
       "                                                            ⎠\n",
       "─────────────────────────────────────────────────────────────\n",
       "                                                             \n",
       "                                                             "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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kriuMb4Et/haQl5ltZ2Yr8MRNDrO9zxpo5MjH6tp2ZIyJ25cWKydJ9+0qfQg5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\nSeBI1p/+BzJWd0nUCOJUAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{\\sqrt{\\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )}}}{16 \\operatorname{a_{i}}^{4}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{4}{\\left (\\tau \\right )}} \\left(8 \\operatorname{a_{i}}^{4}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )} \\frac{d^{4}}{d \\tau^{4}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} - 16 \\operatorname{a_{i}}^{4}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d^{3}}{d \\tau^{3}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} - 12 \\operatorname{a_{i}}^{4}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\left(\\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{j}}{\\left (\\tau \\right )}\\right)^{2} + 36 \\operatorname{a_{i}}^{4}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )}\\right)^{2} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} - 15 \\operatorname{a_{i}}^{4}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )}\\right)^{4} + 8 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{4}{\\left (\\tau \\right )} \\frac{d^{4}}{d \\tau^{4}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} + 16 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )} \\frac{d^{3}}{d \\tau^{3}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} + 16 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d^{3}}{d \\tau^{3}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} + 24 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} - 24 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} - 12 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )}\\right)^{2} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} + 12 \\operatorname{a_{i}}^{3}{\\left (\\tau \\right )} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )}\\right)^{3} - 16 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{4}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )} \\frac{d^{3}}{d \\tau^{3}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} - 12 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{4}{\\left (\\tau \\right )} \\left(\\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{i}}{\\left (\\tau \\right )}\\right)^{2} - 12 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )}\\right)^{2} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{j}}{\\left (\\tau \\right )} - 24 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )} \\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} + 6 \\operatorname{a_{i}}^{2}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{2}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )}\\right)^{2} \\left(\\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )}\\right)^{2} + 36 \\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{4}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )}\\right)^{2} \\frac{d^{2}}{d \\tau^{2}}  \\operatorname{a_{i}}{\\left (\\tau \\right )} + 12 \\operatorname{a_{i}}{\\left (\\tau \\right )} \\operatorname{a_{j}}^{3}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )}\\right)^{3} \\frac{d}{d \\tau} \\operatorname{a_{j}}{\\left (\\tau \\right )} - 15 \\operatorname{a_{j}}^{4}{\\left (\\tau \\right )} \\left(\\frac{d}{d \\tau} \\operatorname{a_{i}}{\\left (\\tau \\right )}\\right)^{4}\\right)$$"
      ],
      "text/plain": [
       "                 ⎛                                                            \n",
       "                 ⎜                   4                                        \n",
       "  ______________ ⎜    4       3     d                 4       2    d          \n",
       "╲╱ aᵢ(τ)⋅a_j(τ) ⋅⎜8⋅aᵢ (τ)⋅a_j (τ)⋅───(a_j(τ)) - 16⋅aᵢ (τ)⋅a_j (τ)⋅──(a_j(τ))⋅\n",
       "                 ⎜                   4                             dτ         \n",
       "                 ⎝                 dτ                                         \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                             2                                \n",
       "  3                             ⎛  2        ⎞                                 \n",
       " d                 4       2    ⎜ d         ⎟         4           ⎛d         ⎞\n",
       "───(a_j(τ)) - 12⋅aᵢ (τ)⋅a_j (τ)⋅⎜───(a_j(τ))⎟  + 36⋅aᵢ (τ)⋅a_j(τ)⋅⎜──(a_j(τ))⎟\n",
       "  3                             ⎜  2        ⎟                     ⎝dτ        ⎠\n",
       "dτ                              ⎝dτ         ⎠                                 \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "2   2                                 4                      4                \n",
       "   d                 4    ⎛d         ⎞        3       4     d                3\n",
       " ⋅───(a_j(τ)) - 15⋅aᵢ (τ)⋅⎜──(a_j(τ))⎟  + 8⋅aᵢ (τ)⋅a_j (τ)⋅───(aᵢ(τ)) + 16⋅aᵢ \n",
       "    2                     ⎝dτ        ⎠                       4                \n",
       "  dτ                                                       dτ                 \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                        3                                          3          \n",
       "       3    d          d                 3       3    d           d           \n",
       "(τ)⋅a_j (τ)⋅──(aᵢ(τ))⋅───(a_j(τ)) + 16⋅aᵢ (τ)⋅a_j (τ)⋅──(a_j(τ))⋅───(aᵢ(τ)) + \n",
       "            dτ          3                             dτ           3          \n",
       "                      dτ                                         dτ           \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                    2          2                                              \n",
       "     3       3     d          d                 3       2    d         d      \n",
       "24⋅aᵢ (τ)⋅a_j (τ)⋅───(aᵢ(τ))⋅───(a_j(τ)) - 24⋅aᵢ (τ)⋅a_j (τ)⋅──(aᵢ(τ))⋅──(a_j(\n",
       "                    2          2                             dτ        dτ     \n",
       "                  dτ         dτ                                               \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "      2                                         2   2                         \n",
       "     d                 3       2    ⎛d         ⎞   d                3         \n",
       "τ))⋅───(a_j(τ)) - 12⋅aᵢ (τ)⋅a_j (τ)⋅⎜──(a_j(τ))⎟ ⋅───(aᵢ(τ)) + 12⋅aᵢ (τ)⋅a_j(τ\n",
       "      2                             ⎝dτ        ⎠    2                         \n",
       "    dτ                                            dτ                          \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                  4       4                                                   \n",
       "             16⋅aᵢ (τ)⋅a_j (τ)                                                \n",
       "\n",
       "                                                                              \n",
       "                        3                                 3                   \n",
       "  d         ⎛d         ⎞         2       4    d          d                2   \n",
       ")⋅──(aᵢ(τ))⋅⎜──(a_j(τ))⎟  - 16⋅aᵢ (τ)⋅a_j (τ)⋅──(aᵢ(τ))⋅───(aᵢ(τ)) - 12⋅aᵢ (τ)\n",
       "  dτ        ⎝dτ        ⎠                      dτ          3                   \n",
       "                                                        dτ                    \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                     2                                                        \n",
       "         ⎛  2       ⎞                                 2   2                   \n",
       "    4    ⎜ d        ⎟         2       3    ⎛d        ⎞   d                 2  \n",
       "⋅a_j (τ)⋅⎜───(aᵢ(τ))⎟  - 12⋅aᵢ (τ)⋅a_j (τ)⋅⎜──(aᵢ(τ))⎟ ⋅───(a_j(τ)) - 24⋅aᵢ (τ\n",
       "         ⎜  2       ⎟                      ⎝dτ       ⎠    2                   \n",
       "         ⎝dτ        ⎠                                   dτ                    \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                 2                                      2     \n",
       "     3    d         d           d               2       2    ⎛d        ⎞  ⎛d  \n",
       ")⋅a_j (τ)⋅──(aᵢ(τ))⋅──(a_j(τ))⋅───(aᵢ(τ)) + 6⋅aᵢ (τ)⋅a_j (τ)⋅⎜──(aᵢ(τ))⎟ ⋅⎜──(\n",
       "          dτ        dτ           2                           ⎝dτ       ⎠  ⎝dτ \n",
       "                               dτ                                             \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "        2                               2   2                                 \n",
       "       ⎞                4    ⎛d        ⎞   d                       3    ⎛d    \n",
       "a_j(τ))⎟  + 36⋅aᵢ(τ)⋅a_j (τ)⋅⎜──(aᵢ(τ))⎟ ⋅───(aᵢ(τ)) + 12⋅aᵢ(τ)⋅a_j (τ)⋅⎜──(aᵢ\n",
       "       ⎠                     ⎝dτ       ⎠    2                           ⎝dτ   \n",
       "                                          dτ                                  \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                           ⎞\n",
       "     3                                    4⎟\n",
       "    ⎞  d                  4    ⎛d        ⎞ ⎟\n",
       "(τ))⎟ ⋅──(a_j(τ)) - 15⋅a_j (τ)⋅⎜──(aᵢ(τ))⎟ ⎟\n",
       "    ⎠  dτ                      ⎝dτ       ⎠ ⎟\n",
       "                                           ⎠\n",
       "────────────────────────────────────────────\n",
       "                                            \n",
       "                                            "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from __future__ import division\n",
    "from sympy import *\n",
    "from IPython.display import display\n",
    "init_printing()\n",
    "tau = symbols('tau', positive = True, real = True)\n",
    "\n",
    "a_i = symbols('a_i', cls = Function)(tau)\n",
    "a_j = symbols('a_j', cls = Function)(tau)\n",
    "\n",
    "for idiff in range(0,5):\n",
    "    display(simplify(diff(sqrt(a_i*a_j),tau,idiff)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
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   "codemirror_mode": {
    "name": "ipython",
    "version": 2
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   "file_extension": ".py",
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   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
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