{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "separated-enclosure",
   "metadata": {},
   "outputs": [],
   "source": [
    "# This short notebook will show how to take symbolic derivatives in Python.\n",
    "# We will require the 'SymPy' module.\n",
    "import sympy as sym"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "precious-accent",
   "metadata": {},
   "outputs": [],
   "source": [
    "# First, let's define a symbol x.\n",
    "x = sym.Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "filled-understanding",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 x^{2} + 3$"
      ],
      "text/plain": [
       "2*x**2 + 3"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Then we can define f in terms of x.\n",
    "f = 2*x**2+3\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "manual-exemption",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2(2)^2 + 3 = 11\n"
     ]
    }
   ],
   "source": [
    "# To evaluate f at a particular value of x, we can use 'subs()'.\n",
    "z = f.subs(x, 2)\n",
    "print('2(2)^2 + 3 =', z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "stopped-minneapolis",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4 x$"
      ],
      "text/plain": [
       "4*x"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# To take the derivative of f with respect to x, we use 'diff()'.\n",
    "dfdx = f.diff(x)\n",
    "dfdx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "durable-valley",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "df/dx at x = 3 is 12\n"
     ]
    }
   ],
   "source": [
    "# Of course, we can now evaluate the derivative at a particular value of x.\n",
    "print('df/dx at x = 3 is', dfdx.subs(x, 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "committed-conditions",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\sin^{y^{2}}{\\left(x \\right)}$"
      ],
      "text/plain": [
       "2*sin(x)**(y**2)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We can make our functions more complicated.  Here's a function of two variables.\n",
    "y = sym.Symbol('y')\n",
    "g = 2*sym.sin(x)**y**2\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "finished-credit",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.00273593133124$"
      ],
      "text/plain": [
       "1.00273593133124"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Here's how you can use 'subs()' to subsitute values for multiple variables.\n",
    "z = g.subs({x:1, y:2})\n",
    "sym.N(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "domestic-triple",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 y^{2} \\sin^{y^{2}}{\\left(x \\right)} \\cos{\\left(x \\right)}}{\\sin{\\left(x \\right)}}$"
      ],
      "text/plain": [
       "2*y**2*sin(x)**(y**2)*cos(x)/sin(x)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Here's the x-derivative of g...\n",
    "g.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "focal-investor",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4 y \\log{\\left(\\sin{\\left(x \\right)} \\right)} \\sin^{y^{2}}{\\left(x \\right)}$"
      ],
      "text/plain": [
       "4*y*log(sin(x))*sin(x)**(y**2)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# and here's the y-derivative.\n",
    "g.diff(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "located-voltage",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Alternatively, you can define Python functions.\n",
    "def fcn(x):\n",
    "    return 2*x**2 + 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "hydraulic-indicator",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2(2)^2 + 3 = 11\n"
     ]
    }
   ],
   "source": [
    "# One advantage of this method is that it is easy to now evaluate the value \n",
    "# of fcn at any x.\n",
    "z = fcn(2)\n",
    "print('2(2)^2 + 3 =', z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "sacred-recipe",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "dg/dx = 4*x\n"
     ]
    }
   ],
   "source": [
    "# The derivatives of these functions are calculated in the way.\n",
    "dgdx = fcn(x).diff(x)\n",
    "print('dg/dx =', dgdx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "photographic-label",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "dg/dx = 4*x\n"
     ]
    }
   ],
   "source": [
    "# Note that there is another way to call the 'diff()' function.\n",
    "dgdx = sym.diff(fcn(x), x)\n",
    "print('dg/dx =', dgdx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "north-animal",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.00273593133124$"
      ],
      "text/plain": [
       "1.00273593133124"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Of course, you can have functions of multiple variables.\n",
    "def gcn(x, y):\n",
    "    return 2*sym.sin(x)**y**2\n",
    "sym.N(gcn(1, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "acknowledged-object",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2*y**2*sin(x)**(y**2)*cos(x)/sin(x), 4*y*log(sin(x))*sin(x)**(y**2))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# You can evaluate the partial derivatives in the way that you'd expect:\n",
    "gcn(x, y).diff(x), gcn(x, y).diff(y)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}