{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "7a365fdd",
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "282b233b",
   "metadata": {},
   "source": [
    "## Derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "77f3c29f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(5., requires_grad=True)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = torch.tensor(5.0, requires_grad=True)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ae84c1f",
   "metadata": {},
   "source": [
    "$x = 5.0$\n",
    "\n",
    "$y = x^2 => f(x) = x^2$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "d40fbe0d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(25., grad_fn=<PowBackward0>)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = x ** 2\n",
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f474ae0",
   "metadata": {},
   "source": [
    "$\\frac{dy}{dx} = 2x$\n",
    "\n",
    "$f'(x=5.0) = 2 * 5.0 = 10$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "dd8db750",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(10.)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.backward()\n",
    "\n",
    "x.grad"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27d0695a",
   "metadata": {},
   "source": [
    "## Partial derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "6abe730e",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = torch.tensor(5.0, requires_grad=True)\n",
    "y = torch.tensor(5.0, requires_grad=True)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "93926cf7",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = x**2 + y**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "4f329c02",
   "metadata": {},
   "outputs": [],
   "source": [
    "f.backward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "d1cb5deb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AddBackward0 at 0x1aefc26ad48>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.grad_fn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "d80038fb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(10.)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.grad # partial derivative wrt x at x = 5 and y =5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "6ab20700",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(10.)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.grad # partial derivative wrt y at x = 5 and y =5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "0d44cfc1",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = torch.tensor(5.0, requires_grad=True)\n",
    "y = torch.tensor(5.0, requires_grad=True)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "9396e8bf",
   "metadata": {},
   "outputs": [],
   "source": [
    "f2 = x**2 * y**2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fdad968c",
   "metadata": {},
   "source": [
    "$f2(x, y) = x^2 . y^2$\n",
    "\n",
    "$\\frac{\\partial f2}{\\partial x} = 2x.y^2$\n",
    "\n",
    "$\\frac{\\partial f2}{\\partial y} = x^2.2y$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "4bd23358",
   "metadata": {},
   "outputs": [],
   "source": [
    "f2.backward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "6b101352",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<MulBackward0 at 0x1aefc470a48>"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2.grad_fn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "4731a225",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(250.)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.grad"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "33fba02c",
   "metadata": {},
   "source": [
    "## Successive Differentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "a3f4c644",
   "metadata": {},
   "outputs": [],
   "source": [
    "from torch.autograd import grad\n",
    "\n",
    "def nth_derivative(f, wrt, n=2):\n",
    "    \n",
    "    for i in range(n):\n",
    "        grads = grad(f, wrt, create_graph=True)[0]\n",
    "        f = grads.sum()\n",
    "        \n",
    "    return grads"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "50612387",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = torch.tensor(5.0, requires_grad=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "064bbc40",
   "metadata": {},
   "source": [
    "$f(x) = x^2 + x^3$\n",
    "\n",
    "$f'(x) = 2x + 3x^2$\n",
    "\n",
    "$f''(x) = 2 + 6x$\n",
    "\n",
    "$f''(x=5) = 2 + 6*5 = 32$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "4bd61626",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(32., grad_fn=<AddBackward0>)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = x**2 + x**3\n",
    "\n",
    "# double derivative\n",
    "nth_derivative(f, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "99b5e49d",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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