SymPy_Examples.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# SymPy Examples\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "# import symbolic capability to Python\n",
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "# print things all pretty\n",
    "from sympy.abc import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "# Need to define variables as symbolic for sympy to use them. \n",
    "x, y= symbols(\"x, y\", real = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$36 x^{3} \\left(3 x^{4} + 5\\right)^{2}$$"
      ],
      "text/plain": [
       "                2\n",
       "    3 ⎛   4    ⎞ \n",
       "36⋅x ⋅⎝3⋅x  + 5⎠ "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff((3*x**4+5)**3,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left ( \\frac{b}{- b^{2} + s^{2}}, \\quad 0, \\quad \\left(\\frac{\\pi}{2} \\geq \\left|{\\arg{\\left (b \\right )} + \\pi}\\right| \\vee \\left|{\\arg{\\left (b \\right )} + \\pi}\\right| < \\frac{\\pi}{2}\\right) \\wedge \\left(\\frac{\\pi}{2} \\geq \\left|{\\arg{\\left (b \\right )}}\\right| \\vee \\left|{\\arg{\\left (b \\right )}}\\right| < \\frac{\\pi}{2}\\right)\\right )$$"
      ],
      "text/plain": [
       "⎛    b         ⎛π                                 π⎞   ⎛π                     \n",
       "⎜─────────, 0, ⎜─ ≥ │arg(b) + π│ ∨ │arg(b) + π│ < ─⎟ ∧ ⎜─ ≥ │arg(b)│ ∨ │arg(b)\n",
       "⎜   2    2     ⎝2                                 2⎠   ⎝2                     \n",
       "⎝- b  + s                                                                     \n",
       "\n",
       "    π⎞⎞\n",
       "│ < ─⎟⎟\n",
       "    2⎠⎟\n",
       "      ⎠"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "laplace_transform(sinh(b*t),t,s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[0;31mSignature:\u001b[0m \u001b[0mlaplace_transform\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
       "\u001b[0;31mDocstring:\u001b[0m\n",
       "Compute the Laplace Transform `F(s)` of `f(t)`,\n",
       "\n",
       ".. math :: F(s) = \\int_0^\\infty e^{-st} f(t) \\mathrm{d}t.\n",
       "\n",
       "For all \"sensible\" functions, this converges absolutely in a\n",
       "half plane  `a < \\operatorname{Re}(s)`.\n",
       "\n",
       "This function returns ``(F, a, cond)``\n",
       "where ``F`` is the Laplace transform of ``f``, `\\operatorname{Re}(s) > a` is the half-plane\n",
       "of convergence, and ``cond`` are auxiliary convergence conditions.\n",
       "\n",
       "If the integral cannot be computed in closed form, this function returns\n",
       "an unevaluated :class:`LaplaceTransform` object.\n",
       "\n",
       "For a description of possible hints, refer to the docstring of\n",
       ":func:`sympy.integrals.transforms.IntegralTransform.doit`. If ``noconds=True``,\n",
       "only `F` will be returned (i.e. not ``cond``, and also not the plane ``a``).\n",
       "\n",
       ">>> from sympy.integrals import laplace_transform\n",
       ">>> from sympy.abc import t, s, a\n",
       ">>> laplace_transform(t**a, t, s)\n",
       "(s**(-a)*gamma(a + 1)/s, 0, -re(a) < 1)\n",
       "\n",
       "See Also\n",
       "========\n",
       "\n",
       "inverse_laplace_transform, mellin_transform, fourier_transform\n",
       "hankel_transform, inverse_hankel_transform\n",
       "\u001b[0;31mFile:\u001b[0m      ~/anaconda3/lib/python3.6/site-packages/sympy/integrals/transforms.py\n",
       "\u001b[0;31mType:\u001b[0m      function\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "laplace_transform?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIwAAAAgBAMAAAAyMI2YAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nkrCVAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$60 x^{2} \\left(5 x^{3} + 2\\right)^{3}$$"
      ],
      "text/plain": [
       "                3\n",
       "    2 ⎛   3    ⎞ \n",
       "60⋅x ⋅⎝5⋅x  + 2⎠ "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff((5*x**3+2)**4,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFQAAAAzBAMAAAD7tDIQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3w4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{7}{\\left(2 x + 1\\right)^{2}}$$"
      ],
      "text/plain": [
       "    7     \n",
       "──────────\n",
       "         2\n",
       "(2⋅x + 1) "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff((3*x-2)/(2*x+1),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGYAAAAzBAMAAACXsmAVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$- \\frac{3}{\\left(2 x + 1\\right)^{2}}$$"
      ],
      "text/plain": [
       "   -3     \n",
       "──────────\n",
       "         2\n",
       "(2⋅x + 1) "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff((1-x)/(2*x+1),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGUAAAAzBAMAAAB8hdsWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nHZFrQrYKUwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{4 x}{\\left(x^{2} - 3\\right)^{2}}$$"
      ],
      "text/plain": [
       "  -4⋅x   \n",
       "─────────\n",
       "        2\n",
       "⎛ 2    ⎞ \n",
       "⎝x  - 3⎠ "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff((0*x-2)/(3-x**2),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF0AAAAzBAMAAAAHoRnaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nylJG95HFE4NkvSBjOZVXAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{12 x}{\\left(2 x^{2} + 1\\right)^{4}}$$"
      ],
      "text/plain": [
       "    12⋅x   \n",
       "───────────\n",
       "          4\n",
       "⎛   2    ⎞ \n",
       "⎝2⋅x  + 1⎠ "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff((-1)/(2*x**2+1)**3,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAA6BAMAAACudAFuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- \\frac{6 \\left(x + 1\\right)^{2}}{\\left(x - 1\\right)^{4}}$$"
      ],
      "text/plain": [
       "          2 \n",
       "-6⋅(x + 1)  \n",
       "────────────\n",
       "         4  \n",
       "  (x - 1)   "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff((x+1)**3/(x-1)**3,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAwBAMAAABJWYGiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8uKy9ZhuZLp2Yh0Ql33sIO3jB8WRoT/ECgnAAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{3}{\\sqrt{6 x - 2}}$$"
      ],
      "text/plain": [
       "     3     \n",
       "───────────\n",
       "  _________\n",
       "╲╱ 6⋅x - 2 "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff(sqrt(6*x-2),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAwBAMAAABJWYGiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJqyLvZs27djJE\nVN3lLQH2AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAB90lEQVQ4Ee2UT0sbURTFzyQZjUkmPFBEughB\n3QgtTdFN3YwIWiiCg8W1ulAqdBEoVOiiKBTBjX6ALoyCtuhC6SeYVRfdGIWuG60LV5og4r9qvDPT\nl7mJzyS7brwweeee++PNmwc5gFNaR4/lioqfka4v3GlN65O8/6e1Ppg28weBftZKGRJoGJcNrReA\nyVopG3IIXcuGjnYJbG37vVThQhmGt7SbrW0M/OyVhFzjBalopTcupZvxKXPETFfupJgTzhhm9gVM\n8YuZrjwsMz5umtsCC66nLR5THblnDSTLMGBYAFcVHrBa6XwGouzjvbGRxCgDP1jRM+xG8lhhJsnv\nwBpzvom2dOymKa+nmQnof7o7c8wJtj+F9npldYZ5JMPFYpFj5dPHro4boBuso+rY6P8jY9U+JF86\n31BJVRP6bLVpaRbI6t0vs6X2IZHACIxz5dSY8O1n2Mri1u991dLp54OWww9bo2BSVMTHgs5f6oGX\nMizh7JJIQZVKDPtK1JsDAVUqeVjMJiRHD4LvoEolD1ufBSIZB8OUUKWSi+lP/gJxYihTlul+76eS\nt5sTvO8JKwoHu59K8DA8t7BP2DRwYilSSWJjycYUYTMwrhSpRAf3rjdUiAvCYu1dtiKVEBi8eEVj\nRE/3nKVmzc/VRBzgd7IuLGzVxO4ApP/BE8LNQ5gAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{9 x - 2}{\\sqrt{3 x - 1}}$$"
      ],
      "text/plain": [
       "  9⋅x - 2  \n",
       "───────────\n",
       "  _________\n",
       "╲╱ 3⋅x - 1 "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff(2*x*sqrt(3*x-1),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGsAAAAwBAMAAADk2JkLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\frac{1}{2 \\sqrt{x}} - \\frac{0.25}{x^{\\frac{3}{2}}}$$"
      ],
      "text/plain": [
       " 1     0.25\n",
       "──── - ────\n",
       "2⋅√x    3/2\n",
       "       x   "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sqrt(x)+1/2/sqrt(x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAUBAMAAAAuMzNdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABKUlEQVQoFX2OPUvDUBiFn0uTfthSQt0cJHTp\nolRwcwpu4nInHf1Al1pQBO0itD/AxVH9CR0q+AOqgWwuig46dXYoFfwIosTcNJEEg3c57zkP77kv\npLxllWlb04MUxmJDpXnvPQ3CU0Dv5X+0kw7D3Rk7fTloFp3yHKI6b6mKoqPNRlUB9a9+YYWcq9KC\n5+0laFbqLttwqtLi4XUEx/8WDPHGGexafp79ZfAIkzIncyZ9+YdWek2WDFq1YKGnLs8ObQv9ZuP4\nKFbij/kP5TNSuKzyfNVN0on10C9wQFveJmk1snVDchmZUEtmMOxAuwNfYRrJHcLw53OoS7TXKB6r\nZlJS1IQuD6URTpxPtfYbrF1QoXyS/86MxCBO+573iX+2qG1KvenYwzj8AZTORUcuWTiYAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$2.5 x^{1.5}$$"
      ],
      "text/plain": [
       "     1.5\n",
       "2.5⋅x   "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff(x**(5/2),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAA5BAMAAAAYbGLhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\frac{2 \\sqrt{t} + 1}{4 \\sqrt{t} \\sqrt{\\sqrt{t} + t}}$$"
      ],
      "text/plain": [
       "    2⋅√t + 1   \n",
       "───────────────\n",
       "       ________\n",
       "4⋅√t⋅╲╱ √t + t "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff(sqrt(t+sqrt(t)),t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARYAAAAWBAMAAAD3FK7IAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$7 x^{6} - 10 x^{4} + 12 x^{3} + 21 x^{2} - 14$$"
      ],
      "text/plain": [
       "   6       4       3       2     \n",
       "7⋅x  - 10⋅x  + 12⋅x  + 21⋅x  - 14"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff((3-2*x+x**3)*(x**4+7),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANEAAAAWBAMAAACs+WTQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngEx2g/p/0mOrksDOgUAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$5 t^{4} + 4 t^{3} + 3 t^{2} + 2 t + 1$$"
      ],
      "text/plain": [
       "   4      3      2          \n",
       "5⋅t  + 4⋅t  + 3⋅t  + 2⋅t + 1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff((t**3+1)*(t**2+t+1),t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjMAAAAgBAMAAAAVoeTtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMol2RJm7Iu9m\nVKtxfBG3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAH1klEQVRoBe1YbYhUVRh+787nzsd6MfSHgjut\nZfQj3SxDLGr6IVlRO1m7bBQ6BsqSoBP0IUTtrLSyS7KNEJREORCihbBagpSIA2FgSjtClBSxI4WU\nH7hrZSLp9J5z7rnnPXfOvbs72L/uj7nv53Oe973nvPfuAtzM68nHTjYJ1zNiN5kZmNYa6A1yloKc\nTfj667UmsjAlXonfbC6cyHizDU9PNleIb9YbD/u6gh3xXGoiOKIpr7W22dbMudmtea2pAnjSf7Jr\nou3NtmZdYGtKPnUGjIUtGwZ8kpg5cBLNzwZkoqv3fZ8qI31nfDOfCGpN+Nd9Fb/M8Kmg1vidtqCx\nsAeO+y2G9qBJFF0UkIiuSLktY47YDsNmB1pzQa2J1W/4JkasoNbI0xYu6AA+YyEvorrK/P6I0PRf\n8ySKiqApDlSoHL4mAr18BmFMX4ZrHDVVlq0x8Yl9Z9yHeZb/TWBr5Gl7lC9Ff4xVvMkizkA7b2S4\nROOlbJ5EsRr3J67LMOM9VEj8LRyNfEy7JlbD6LngtMbIJ+Z55gIeWCFWPqg17mm71UlRN+NYiFQw\n4EsY5U+ihf+qDCH5TKLn0Ts/m/jHG+7R5QFv4GMd9ERylaGeHnpnO1eMfGLb+kw0WSHpoaGrePO5\n5GmL570BPmPhbox7Cnp59DZvDtd9JlEICYbK0QPGHGUM5bncyOcF4eBe9cNQwXlSYOSTKKRKKl5J\nrBBI0FkT6XkGrMFjWSdGnjaxDRyjuHkPVGJoJ2b1oTPSm2MhFDfR/WyV2fjlTCKpinu0CBDuWVPW\nrVxbZK9yk0XTZbE02Hug9g6/C8BQIbXrIgukfHZ3f8hM/JLjS+rizgqB4zfchcG6A7ZCS9VaIfzu\naXMYCav49Y6FBRDBOTCqQiJFJe+FtDwqchIpJ/xkI/EJYlAicx2t/yUNqaKQDHxGVRUsxqrB2oKG\nSvgkijBeEUhttuWML6Hjb5ptXVKIcLQWYSecAFgtVPe0HRG6+lVjIZ0X1jMAh/FMVNyYJOG6uQp/\nOA45iQC+Fxbrt0tYPzzgJipBuG4/W5amV+BlLnr54KQZrYggB7XNhuR+DZXwiV6HWZ0iPFl2d5OT\nuWCEHSNSiAjsL+P9I4BxRpZdIi9Mzxy3q7EgW/M5wKUskGPWJTEw4fWCdZnnqUnktgZgM4v0HggR\nzlw1JxNP3F0/72BKI58VsNFZzikw2Qltf2qohE/sELRnBGrEdneTkwlRVi0pRAR+wG438LusKnTn\ntKX2O6p7U2NBtmaJzVoTVZG4+cjlHihnEqFLUhGt2UiClai3JlmvTzBfI5+V6885SQ5qyyRvDUHV\n+cgDBd2DcjXJh7eGFCL8l1f2Fayr2Jq8teHcqz0yKZ6REt7nbtqzPq902Rq0PGRDTO2vrSoGpVty\noEGiSVIRrem3tXhHYa258PVbnlUpH19UCCEXgqrxsZZ5IAkf3hpSCKdiXa7AkQQOva7cHPgqv1SS\nbS1JCedbLnVfKKN01ZoYJoZxEzvXTimw++7lNmiQaNNb05Vlcd6LtWYHjBb0VSkfX1Toz2EhWReR\n8kk8PdBQiMuHt4YUwhGsug1vz8Nd05V7CtbaP0rYlpqU8NVsRybTZaWr1oQwSn6ton+xikEpcgi/\ndwgkmvTWjBa0eEdhrQFo7dRXpXx8UeFBzCSoOp/nCjokxko+vDVuIbPvYddt7DUy9hI/UDbcyUiJ\nK1TE3bKQhSzNWtBSknaIdHQs/Kyjo8YNQ/hLXunLUZW4KB62NcjVHR33d3QIurx+9fKVKzFQ0Zr4\nNW1VYHwktC9qHINAoQLyIcjJAzok4SNa4/2TBY/QWAXH8HgV4AoiOxd9SgCz8tLO7u6u4Z9XbrP1\nXfMpwC4dEjPlUxL1k+dL4LE1+Mc2+2Kiq+p8CFGKyj/aCCrZNYkytGLpFJJk6rtGkjmPrSngoRyz\n6dQAnQo5v5jntuY04Irk05JQATyo2BrP+dVbY/xGZl1LZiCOA5WuqvHxQU0XAR8IQSV8Zk3y1lBI\nb2tIIaI5XThr2CffVjgVnYB1smNt6ghB2t4MUVt6VGvCRYiW6RvqdxUDX+Drq6xBolNvDXmXkERs\nTRy/vzL6qpSPH+oqgBe1NxThkyxB6JoOSfjwXeN9Q0FbzjoCoap1MHa9dSJRkRzpS76/ugv2SQfe\n5a75uLv3F+2Lg3xSwElIX9EhMVNvjf7ZIRfA1iSKcKKsr0r4+KEmlnWPdAJ7zPIifFIVGM/rkBgl\n+fDWNH469Y5kwerZlLXOrus7KUHpXoAFG+Zur7ge1Zol9ToO8bjaX3QbxIaGCzokoTJydDECDhNM\nV+SuvWsGPKuSR+qH2lKv17E1BJXy2TL0ngdS8Ymfv3xBK8RlYxQOGa3cKHeNiGjJu5HJgiuaBfmU\nhHeROchsDeDjizozPqQQMwVpHZRC490qU1u77WrxnCuahXnUHPQPNRon5AA+vqgz40MKaVyeWh6n\nSpD8rXKG9yt5ailVmzpGRUyXD0WdGR9SiFrWJLVVTVaD7V5iu0jkKcVQdsoQEjBdPhrqjPjQQsjC\njaKVabSZLCl6iGabIvxsx/wcRruVMZobjBrqTPhohTTAaobTmuarfEI96SLVguVEKdjv9U6Pj446\nEz5aId7FdT1d0HUfbUCz/6BpgUrcDnQ3OKfHx4M6Az56IQ3L/2/ADvwLf6QeS7K3Ao8AAAAASUVO\nRK5CYII=\n",
      "text/latex": [
       "$$- 60 x^{4} \\left(- x^{3} + 2\\right)^{5} \\left(- 3 x^{5} + 1\\right)^{3} - 15 x^{2} \\left(- x^{3} + 2\\right)^{4} \\left(- 3 x^{5} + 1\\right)^{4}$$"
      ],
      "text/plain": [
       "                  5             3                   4             4\n",
       "      4 ⎛   3    ⎞  ⎛     5    ⎞        2 ⎛   3    ⎞  ⎛     5    ⎞ \n",
       "- 60⋅x ⋅⎝- x  + 2⎠ ⋅⎝- 3⋅x  + 1⎠  - 15⋅x ⋅⎝- x  + 2⎠ ⋅⎝- 3⋅x  + 1⎠ "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(diff((-3*x**5+1)**4*(2-x**3)**5,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAzBAMAAAAk+kgPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQjouUfIpJOAPxHWwAd41yaUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{20 t}{\\left(t^{2} + 5\\right)^{2}}$$"
      ],
      "text/plain": [
       "   20⋅t  \n",
       "─────────\n",
       "        2\n",
       "⎛ 2    ⎞ \n",
       "⎝t  + 5⎠ "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff((t**2-5)/(t**2+5),t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALcAAAAxBAMAAABqqSVpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\frac{x^{4} + 14 x^{2} - 10 x - 8}{x^{4} + 8 x^{2} + 16}$$"
      ],
      "text/plain": [
       " 4       2           \n",
       "x  + 14⋅x  - 10⋅x - 8\n",
       "─────────────────────\n",
       "     4      2        \n",
       "    x  + 8⋅x  + 16   "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff((x**3-2*x+5)/(x**2+4),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALsAAAA6BAMAAAAaUjUkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\left(2 x - 7\\right)^{2}}{\\left(x - 1\\right)^{6}} \\left(- 4 x + 29\\right)$$"
      ],
      "text/plain": [
       "                     2\n",
       "(-4⋅x + 29)⋅(2⋅x - 7) \n",
       "──────────────────────\n",
       "              6       \n",
       "       (x - 1)        "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff((2*x-7)**3/(x-1)**5,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAAA6BAMAAABmGOu4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nP/44+fXRj/0VAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{12 \\left(2 t + 1\\right)^{2}}{\\left(2 t - 1\\right)^{4}}$$"
      ],
      "text/plain": [
       "             2 \n",
       "-12⋅(2⋅t + 1)  \n",
       "───────────────\n",
       "            4  \n",
       "   (2⋅t - 1)   "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff(((2*t+1)/(2*t-1))**3,t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJUAAAA3BAMAAAD3WBWZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{495 x^{4} - 66}{\\left(3 x^{5} - 2 x + 7\\right)^{12}}$$"
      ],
      "text/plain": [
       "     ⎛    4    ⎞  \n",
       "  33⋅⎝15⋅x  - 2⎠  \n",
       "──────────────────\n",
       "                12\n",
       "⎛   5          ⎞  \n",
       "⎝3⋅x  - 2⋅x + 7⎠  "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = simplify(diff((-3)/(3*x**5-2*x+7)**11,x))\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a == 33*(3*x**5-2*x+7)*(15*x**4-2)/(3*x**5-2*x+7)**22"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "but that was way too easy. \n",
    "\n",
    "We could compare them, just to be sure they are the same"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff(y,x))==simplify(diff(ln(y),x)*y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Or plot this monstrosity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jslater/Library/Python/3.5/lib/python/site-packages/sympy/plotting/experimental_lambdify.py:230: UserWarning: The evaluation of the expression is problematic. We are trying a failback method that may still work. Please report this as a bug.\n",
      "  warnings.warn('The evaluation of the expression is'\n"
     ]
    },
    {
     "ename": "AttributeError",
     "evalue": "'Symbol' object has no attribute 'imag'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m/Users/jslater/Library/Python/3.5/lib/python/site-packages/sympy/plotting/experimental_lambdify.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, args)\u001b[0m\n\u001b[1;32m    193\u001b[0m             \u001b[0;31m#The result can be sympy.Float. Hence wrap it with complex type.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 194\u001b[0;31m             \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcomplex\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlambda_func\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    195\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mimag\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m1e-7\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/jslater/Library/Python/3.5/lib/python/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m__complex__\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    230\u001b[0m         \u001b[0mre\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mim\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_real_imag\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 231\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mcomplex\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mre\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mim\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    232\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/jslater/Library/Python/3.5/lib/python/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m__float__\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    225\u001b[0m             \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"can't convert complex to float\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 226\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"can't convert expression to float\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    227\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: can't convert expression to float",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-12-2321e2b00f70>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdiff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m0.2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m10\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m0.5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m10\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/Users/jslater/Library/Python/3.5/lib/python/site-packages/sympy/plotting/plot.py\u001b[0m in \u001b[0;36mplot\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m   1282\u001b[0m     \u001b[0mplots\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPlot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mseries\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1283\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mshow\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1284\u001b[0;31m         \u001b[0mplots\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1285\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mplots\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1286\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/jslater/Library/Python/3.5/lib/python/site-packages/sympy/plotting/plot.py\u001b[0m in \u001b[0;36mshow\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    183\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_backend\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mclose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    184\u001b[0m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_backend\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbackend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 185\u001b[0;31m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_backend\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    186\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    187\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0msave\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpath\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/jslater/Library/Python/3.5/lib/python/site-packages/sympy/plotting/plot.py\u001b[0m in \u001b[0;36mshow\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m   1016\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1017\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1018\u001b[0;31m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mprocess_series\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1019\u001b[0m         \u001b[0;31m#TODO after fixing https://github.com/ipython/ipython/issues/1255\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1020\u001b[0m         \u001b[0;31m# you can uncomment the next line and remove the pyplot.show() call\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/jslater/Library/Python/3.5/lib/python/site-packages/sympy/plotting/plot.py\u001b[0m in \u001b[0;36mprocess_series\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    895\u001b[0m             \u001b[0;31m# Create the collections\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    896\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_2Dline\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 897\u001b[0;31m                 \u001b[0mcollection\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mLineCollection\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_segments\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    898\u001b[0m                 \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0max\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0madd_collection\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcollection\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    899\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_contour\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/jslater/Library/Python/3.5/lib/python/site-packages/sympy/plotting/plot.py\u001b[0m in \u001b[0;36mget_segments\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    501\u001b[0m                     \u001b[0mlist_segments\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mq\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    502\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 503\u001b[0;31m             \u001b[0mf_start\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstart\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    504\u001b[0m             \u001b[0mf_end\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mend\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    505\u001b[0m             \u001b[0msample\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstart\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_start\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mend\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_end\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/jslater/Library/Python/3.5/lib/python/site-packages/sympy/plotting/experimental_lambdify.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, args)\u001b[0m\n\u001b[1;32m    231\u001b[0m                         \u001b[0;34m' problematic. We are trying a failback method'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    232\u001b[0m                         ' that may still work. Please report this as a bug.')\n\u001b[0;32m--> 233\u001b[0;31m                 \u001b[0;32mif\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mimag\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m1e-7\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    234\u001b[0m                     \u001b[0;32mreturn\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    235\u001b[0m                 \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mAttributeError\u001b[0m: 'Symbol' object has no attribute 'imag'"
     ]
    }
   ],
   "source": [
    "plot((diff(y,x),(x,0.2,10)), (y, (x,0.5,10)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    },
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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JyFoIIbDuyDpM2zoN209sR/dW3REXEoehnYZytjG6KY4GJyIysXJRjtWHViP6z2gczTsK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\nEs1IwQj0KY8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b7b2a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# To change colors\n",
    "# show = False delays the plot until we can set all of the parameters\n",
    "# legend turns on the legend and uses the labels we have later. \n",
    "p = plot((diff(y,x),(x,0.2,10)), (y, (x,0.5,10)), show = False, legend = True)\n",
    "p[0].line_color = 'blue'\n",
    "p[0].label = '$\\\\frac{dy}{dx}$'\n",
    "p[1].line_color = 'green'\n",
    "p[1].label = '$y$'\n",
    "p.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "What if we wanted to make a substitution?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\frac{\\alpha^{2} \\sqrt{3 \\alpha - 2}}{\\left(\\alpha + 1\\right)^{2}}$$"
      ],
      "text/plain": [
       " 2   _________\n",
       "α ⋅╲╱ 3⋅α - 2 \n",
       "──────────────\n",
       "          2   \n",
       "   (α + 1)    "
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.subs(x,alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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    "deletable": true,
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    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": []
  }
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