MathWithSumpy_DERIVATiVES_082918.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Just about any math equation !!!\n",
    " \n",
    " - Derivative of one dimensional variable\n",
    " - Derivative of multimensional variable\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x, y, z = symbols(\"x, y, z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a, b,c,d = symbols ('a b c d')\n",
    "f,g = symbols('f g', cls=Function)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simple One Dimensional variable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAAVBAMAAACj2UNSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMpndu3bvImbNiRBU\nq0Qb3U6NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuUlEQVQYGWNgQAJLdTcg8RiYGpgcUPgLOD4g\n8xkYmBxQ+dsEUPisU8FcIbggkwOIaQIiwID9D5BiTYHytgmw/wAyK6ZA+cwFrNlA5gIYn3Hp2gIG\nBo4CIH9rbF1UAETZVoYpDNwLOEyZJ0D4C4B8rg1cv3mAaoGARwAkz8DpAOYxMOxiAPIZGPihuhnO\n3r377SoDg7wAVB5IpTDwbIhnYN0AE/nC8P6AP0MFjLvk/429oVs1GmB8FBoAhj0pVZCY1KcAAAAA\nSUVORK5CYII=\n",
      "text/latex": [
       "$$4 x^{3}$$"
      ],
      "text/plain": [
       "   3\n",
       "4⋅x "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**4, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAVBAMAAAA3P+qgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiuzKJ\nRN0MreaOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA10lEQVQYGWNgQAM8UcFoIkDuSQZ3TMGXDPsn\nYIi+ZZh/AUOQgUHeAIvgM6gYZ+iqDTBpngAgi1GZgcGEgecvTLAVyDAJ+czA4MzAoAQV5D3AC9TE\nBhTUY2Dwh+rfLihlABGsN4AL6v//D9QDUgkE7w2Y0zyvHQOzYYK8fxhsGPoSnqMI8jkwHGY4byCJ\nIhjKwGDAUAUVgmpnA7mY4RuqoCgD5wQG7t8ogtwBDGwThNk+MCRDhUFOmnfmqB/vR64PnA0QQVan\n7x4M9f///2N2S07tgOvHZAAAvbA5b/bh4YYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$12 x^{2}$$"
      ],
      "text/plain": [
       "    2\n",
       "12⋅x "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**4, x,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAAPBAMAAAABlGKvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvElEQVQIHVXOwQoBYRDA8f9mt2VtcVZK+wRb\nHoBH2BNXDhykVkouygs4K17BwcED2JSDciHuHFwoOQiJmE/twRxm5jdN06A52TygZySpKGA+pFju\nT1CBgbSL0EPw82in0FNP2YyHlt2xx15sLMu9jjB6xwjERY7BSGyVMBG36HorsQMbZY+JCDuD0VCG\nl/IaLV2t+v0A/SqUR+0kxFy29oU5qXazJuOEG31HLtqO6efzBDN3O9Tns7M68BdfzNEtOQyaZk4A\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$24 x$$"
      ],
      "text/plain": [
       "24⋅x"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**4, x,x,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAPBAMAAAD0aukfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAhElEQVQIHWNgVDJ2YGBgYFFgYAhjYP8BZHIZ\nMDCkMjDMBDJPApmzGBjyHRgYXwOZ+wNATHZuIBMI1gcwPIQwOb4zsB6AMLkSGNgZIEwlBoYrECaP\nAgNrAYR5iYFRNi0tf9oBkI08AgwMnEDDpMqLsoAG8gOZ+////8XAwG7/7QHICigAACxIGvLdVPLy\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$24$$"
      ],
      "text/plain": [
       "24"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**4, x,x,x,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAN0AAAAcBAMAAAAXXqc4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/gUBVerOKugjfAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left(x^{2} y^{2} z^{2} + 3 x y z + 1\\right) e^{x y z}$$"
      ],
      "text/plain": [
       "⎛ 2  2  2              ⎞  x⋅y⋅z\n",
       "⎝x ⋅y ⋅z  + 3⋅x⋅y⋅z + 1⎠⋅ℯ     "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# When would this be useful?\n",
    "leeExpr.diff(x,y,z,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### More Complex Derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI8AAAAUBAMAAABG7HMgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\naZy/CP53Pcd/XbV/ATjyeFbUbpJHAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x \\left(x + 1\\right) \\left(x + 2\\right)$$"
      ],
      "text/plain": [
       "x⋅(x + 1)⋅(x + 2)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "leeExpr2 = x*(x+1)*(x+2)\n",
    "leeExpr2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVEAAAAUBAMAAADGuhMaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADLElEQVRIDY2WvWsVQRTFzz437yOJuiQipJGX\nCHYhKilsJNtIAmmClZ0xhWDxMEVALYSH2GihgSi4KCSKhY3yCsHCwvcfJCBoFbARtNEoKCJCvPNx\nZ2dms5mdYvfOnT2/c7K787KAGbOm8opG6jXyabyc12614k7FLGhQkRb3imzdmS5dmd9rJd5+28eB\n1F8KG1SktQpoafUqAcZ8UzN/YCqrGEX9F6JVqyPLsEFF2k2HPNiW0+jsDCWtqwnwzLkGaGzZDa3B\nKeAbsKSXuAvXQKNsgyo0Ab1tm8Lwpyhp9Emv+UmbqS1izRdgpo9jeom7noFB5QZVaASNu3TIh+EL\nEB7qBYPX87lcQBVr1hORtLWgFrkbd9VcHw0qNwjRRo6fO9EGavQgo4np80uKxHxIUMfDaw0mqc91\nnpSa9xPUCSgGkzwDN6k0CNCixcat5iowtAwcxfX2HZevkt5QTX5PWYM39NdpvZUJGPhN7zBBxeCk\nnoGbVBoEaPWk/mMwBZobwCVcTZ4Les5XSV+oJidlDe7SbtN6W4PmSSJ0lYiTegZuUmkQoEVo9QTy\n0AKQ4LGk17Ps0dcsIzuopJuiupJl97JsjSrW4HNe2xps00XDO3SwurZBjrINgrTDbULKpMBPUYrB\nd8JKSl1zI5RGJAV0bWnEK6mS0plJIqllYFByI8hbEaRt9gVCPBwM/xOlGMxXSS+oZp5UacTTB3Rt\naV4iTjHQVSImeQZuUmkQoA0mU6glakc9re1g3OWrpN6OYo3YUabOkw5voZbuuaMsAzcp76j9aBc3\n3slPB/pRGfg7tMMfCXwnVNIPKj7fU9agA5g6Tzp3uXMGOLioREzyDNyk0iBAG50Y+bhMTHpY0enx\nyWs6E/Pff1+j1Se6q/GsAT00U+dJ13d3/wCtvhIxyTPgpJZBFZpglvw3FUtxlw5iMF7N6OVOuRJn\nzqR6/B/HdF0DBxV3haYKTVznfkBEqejpYb5QjnBHn91vCkdjvlBM1zVwUMqgCk0YD6XavngaK7Z0\n53XpSvGrL2xQkRb3Sl3LCfOlmmKusEFV2myZayMtWwH/VBQvWCm2ggb70v4Dernyx7GAvGgAAAAA\nSUVORK5CYII=\n",
      "text/latex": [
       "$$x \\left(x + 1\\right) + x \\left(x + 2\\right) + \\left(x + 1\\right) \\left(x + 2\\right)$$"
      ],
      "text/plain": [
       "x⋅(x + 1) + x⋅(x + 2) + (x + 1)⋅(x + 2)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# First Derivative\n",
    "leeFirstDeriv = leeExpr2.diff(x)\n",
    "leeFirstDeriv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAQBAMAAABelcpIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyEM1Embsi72ZU\n3au6f2Q3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBElEQVQYGWNgYBBUNGDAAhiFhIGirgnsAVgk\nWcIYSoHC7QwcDciyhyAcjgKGDgYGtp/IUkA2VDZ/AkicYwGIRAJQ2S6wEH+wUBkDg5PeNEUFiBKo\n7FdD9QQGBv9oBq4DLALcl3ihboPIsnzdwLAGKPuFgX0JowPjFx6wRTB7Wf47MFQ5MPBfYOD+xsLA\n2gA2VzQ09FZoaByQ/ZeBYX8CA9BA7t8MDPwKYFkgAbX3KlB2AwPfAqBeBgZ7AzTZV2C9zEB7F/A4\n6DMwOUDkoXrtgfYCBR4yeB3IP9DPMBsiCTOZT4AF6GYGxvLHDH5KTtUbUGUZxOvglkFlQBTUZCQR\nZKYnEgcA9fM7LCV28hQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$6 x + 6$$"
      ],
      "text/plain": [
       "6⋅x + 6"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Second Derivative\n",
    "leeSecondDeriv = diff(leeFirstDeriv, n=2)\n",
    "leeSecondDeriv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAAAZBAMAAAAF/IdZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\sin{\\left (y \\right )} \\cos^{2}{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "          2   \n",
       "sin(y)⋅cos (x)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "leeFunc_x_y = sin(y)*cos(x)**2\n",
    "leeFunc_x_y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM4AAAAUBAMAAAA3lB++AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- 2 \\sin{\\left (x \\right )} \\sin{\\left (y \\right )} \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-2⋅sin(x)⋅sin(y)⋅cos(x)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Differentiate with respect to x\n",
    "deriv_leeFunc_x_y = diff(leeFunc_x_y,x)\n",
    "deriv_leeFunc_x_y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAZBAMAAAALeTtzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nZvQE/Zl0Qo12jEedgO5wTLD1v7T/5eyTSdZ54iwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\cos^{2}{\\left (x \\right )} \\cos{\\left (y \\right )}$$"
      ],
      "text/plain": [
       "   2          \n",
       "cos (x)⋅cos(y)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Differentiate with respect to y\n",
    "deriv_leeFunc_x_y = diff(leeFunc_x_y,y)\n",
    "deriv_leeFunc_x_y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPwAAAAcBAMAAAC3wc/WAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n1gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2 \\left(\\sin^{2}{\\left (x \\right )} - \\cos^{2}{\\left (x \\right )}\\right) \\cos{\\left (y \\right )}$$"
      ],
      "text/plain": [
       "  ⎛   2         2   ⎞       \n",
       "2⋅⎝sin (x) - cos (x)⎠⋅cos(y)"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Differentiate with respect to x THEN differentiate with respect to y\n",
    "# THEN differentiate with respect to x again\n",
    "deriv_leeFunc_x_y = diff(leeFunc_x_y,x,y,x)\n",
    "deriv_leeFunc_x_y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKkAAAArBAMAAADxg7XPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nPffZVI+Tf7C82Efq1/+UAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{d}{d x} \\left ( x^{2} \\sin{\\left (x \\right )}, \\quad x\\right )$$"
      ],
      "text/plain": [
       "d ⎛⎛ 2          ⎞⎞\n",
       "──⎝⎝x ⋅sin(x), x⎠⎠\n",
       "dx                "
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Chain rule\n",
    "f = x**2\n",
    "g = sin(x)\n",
    "chainDiff = diff((f*g, x))\n",
    "chainDiff"
   ]
  }
 ],
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  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [conda env:Anaconda3]",
   "language": "python",
   "name": "conda-env-Anaconda3-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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 },
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}