NNFS/07_nnfs_derivatives.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/theaveas/DeepLearning/blob/main/NNFS/07_nnfs_derivatives.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "9c927d65-9642-4a5e-bf92-988eac5d5dfa",
      "metadata": {
        "id": "9c927d65-9642-4a5e-bf92-988eac5d5dfa"
      },
      "source": [
        "# Derivatives\n",
        "## The impact of a Parameter on the Output"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "id": "de464442-b140-49fa-a7be-a52ab430dd8c",
      "metadata": {
        "id": "de464442-b140-49fa-a7be-a52ab430dd8c",
        "outputId": "b2cbdea2-03cd-4e57-9e2f-d1688934d9d1",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "3.7.12\n",
            "1.19.5\n",
            "3.2.2\n"
          ]
        }
      ],
      "source": [
        "import platform\n",
        "import numpy as np\n",
        "import matplotlib\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "print(platform.python_version())\n",
        "print(np.__version__)\n",
        "print(matplotlib. __version__)\n",
        "\n",
        "#python version 3.9.7\n",
        "#numpy version 1.21.2\n",
        "#matplotlib version 3.5.0"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "id": "8f5f14f4-801a-4af6-a2b0-c250ca692003",
      "metadata": {
        "id": "8f5f14f4-801a-4af6-a2b0-c250ca692003",
        "outputId": "8b45c686-4499-430a-f100-41db9e13b84a",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 300
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[0 1 2 3 4]\n",
            "[0 2 4 6 8]\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "# linear function\n",
        "def f(x):\n",
        "    return 2 * x\n",
        "\n",
        "x = np.array(range(5))\n",
        "y = f(x)\n",
        "\n",
        "print(x)\n",
        "print(y)\n",
        "\n",
        "plt.plot(x, y)\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "eaa98fff-6be2-4e17-94a7-b6f2240edf3c",
      "metadata": {
        "id": "eaa98fff-6be2-4e17-94a7-b6f2240edf3c"
      },
      "source": [
        "## The Slope\n",
        "From the function above, some will say, `y is double x` Another way to describe the impact of a linear function such as this comes from algebra: the **slope**\\\n",
        "`change in y` / `change in x` = `dy`/`dx`\\\n",
        "If is change in **y** divideed by change in **x**, to calculate the slope\n",
        "1. take any two points lying on the function's graph \n",
        "2. subtract them to calculating the change\n",
        "3. subtracting the points means to subtract their x and y dimensions respectively"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "id": "add91f57-cda3-4730-ade0-b6b166e6ef78",
      "metadata": {
        "id": "add91f57-cda3-4730-ade0-b6b166e6ef78",
        "outputId": "b99ea9d5-4d9f-4d79-8f67-9effbcff07c6",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "2.0\n"
          ]
        }
      ],
      "source": [
        "# note y = f(x) where x = 2\n",
        "print((y[1]-y[0])/(x[1] - x[0]))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "id": "b343d9a7-3edd-4620-983b-1baa47af9c84",
      "metadata": {
        "id": "b343d9a7-3edd-4620-983b-1baa47af9c84",
        "outputId": "a99351b9-cfd9-4e00-ee3d-e7af44b5d97e",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 300
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[0 1 2 3 4]\n",
            "[ 0  2  8 18 32]\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "# nonlinear function (parabolic)\n",
        "def f(x):\n",
        "    return 2 * x**2\n",
        "\n",
        "x = np.array(range(5))\n",
        "y = f(x)\n",
        "\n",
        "print(x)\n",
        "print(y)\n",
        "\n",
        "plt.plot(x, y)\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "fb6e994c-829a-42c5-87ac-3c425afa6008",
      "metadata": {
        "id": "fb6e994c-829a-42c5-87ac-3c425afa6008"
      },
      "source": [
        "How might we measure the impact theat x has on y in *nonlinear* function? **Calculus** proposes that we measure the slope of the **tangent line**, giving us the **instantaneous slope**, which is the **deravative**(slope).\\\n",
        "The **tangent line** is created by drawing a line between two points that are `infinitely close` on a curve, but this curve has to be differentiable at the derivation point."
      ]
    },
    {
      "cell_type": "markdown",
      "id": "3a5e043a-3e16-42c2-bf68-660bf061336d",
      "metadata": {
        "id": "3a5e043a-3e16-42c2-bf68-660bf061336d"
      },
      "source": [
        "## The Numerical Derivative\n",
        "Calculating the slope of the tangent line using two infinitely close points, or calculating the slope of a tengent line made from two points that were `sufficiently close`. "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "id": "67d96fb8-62b9-48a0-8b6a-30d0bcd89a12",
      "metadata": {
        "id": "67d96fb8-62b9-48a0-8b6a-30d0bcd89a12",
        "outputId": "1977e723-d484-404a-d223-630ae3fff261",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        }
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f570784f310>]"
            ]
          },
          "metadata": {},
          "execution_count": 5
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "def f(x):\n",
        "    return 2*x**2\n",
        "\n",
        "# np.arange(start, stop, step) to give us smoother line\n",
        "x = np.arange(0, 5, 0.001)\n",
        "y = f(x)\n",
        "\n",
        "plt.plot(x, y)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "id": "a35614b9-9470-42dd-a364-c3e43afabb19",
      "metadata": {
        "id": "a35614b9-9470-42dd-a364-c3e43afabb19",
        "outputId": "31fba78e-f995-43de-d664-55f898b9248e",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "(2, 8) (2.0001, 8.000800020000002)\n",
            "8.000199999998785\n"
          ]
        }
      ],
      "source": [
        "# the point and the close enough point\n",
        "p2_delta = 0.0001\n",
        "x1 = 2\n",
        "x2 = 2 + p2_delta\n",
        "\n",
        "y1 = f(x1)\n",
        "y2 = f(x2)\n",
        "\n",
        "print((x1, y1,), (x2, y2))\n",
        "\n",
        "# deraviative approx and y-intercept for the tengent line\n",
        "dapprox = (y2-y1) / (x2-x1)\n",
        "b = y2 - dapprox*x2\n",
        "print(dapprox)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "id": "92940b33-c7b5-42e5-9b29-2398684e8bde",
      "metadata": {
        "id": "92940b33-c7b5-42e5-9b29-2398684e8bde",
        "outputId": "643f2ac7-ca6c-462b-ac69-e03405d0c9cc",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Approximate derivative for f(x) where x = 2 is 8.000199999998785\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "# tengent_line function\n",
        "def tangent_line(x):\n",
        "    return dapprox * x + b\n",
        "\n",
        "# plotting the tengent line\n",
        "to_plot = [x1-0.9, x1+0.9]\n",
        "plt.plot(x, y)\n",
        "plt.plot(to_plot, [tangent_line(i) for i in to_plot])\n",
        "\n",
        "print('Approximate derivative for f(x)', \n",
        "      f'where x = {x1} is {dapprox}')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "b3bbf8bb-e766-4060-9476-a3e21351339f",
      "metadata": {
        "id": "b3bbf8bb-e766-4060-9476-a3e21351339f"
      },
      "source": [
        "## Summary\n",
        "The derivative of a constant equals 0 (m is a constant in this case, as it's not a prameter that we are deriving with repect to , which is x in this example): `dm` / `dx` = `0`\\\n",
        "\\\n",
        "The derivative of x equal 1: `dx` / `dx` = `1`\\\n",
        "\\\n",
        "The derivative of a linear function equals its slope: `dmx + b` / `dx` = `m`\\\n",
        "\\\n",
        "The derivative of a constant multiple of the function equals the constant multiple of the function's derivative: `d[k * f(x)]` / `dx` = `k * df(x)` / `dx`\\\n",
        "\\\n",
        "The derivative of a sum of functions equals the sum of theor derivatives: `d[f(x) + g(x)]` / `dx` = `df(x)` / `dx`  + `d(gx)` / `dx`\\\n",
        "\\\n",
        "The derivative of an exponentiation: `dx**n` / `dx` = `n` * `x**n-1`"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "0ec10450-165e-451e-9999-0220c5982020",
      "metadata": {
        "id": "0ec10450-165e-451e-9999-0220c5982020"
      },
      "source": [
        "[**Learn more about Calculus**](https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr) A Youtube playlist by **3blue1brown**"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "torch110",
      "language": "python",
      "name": "torch110"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.9.7"
    },
    "colab": {
      "name": "07_nnfs_derivatives.ipynb",
      "provenance": [],
      "include_colab_link": true
    }
  },
  "nbformat": 4,
  "nbformat_minor": 5
}