{ "cells": [ { "cell_type": "markdown", "id": "8fef0a39", "metadata": {}, "source": [ "要对一个函数求导，需要安装sympy库。\n", "pip install sympy" ] }, { "cell_type": "code", "execution_count": 30, "id": "701e82ca", "metadata": {}, "outputs": [], "source": [ "from sympy import diff, symbols, exp\n", "import numpy as np\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 12, "id": "9e0c42d5", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = np.arange(0, 5)\n", "y = [1] * len(x)\n", "plt.plot(x, y)\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "6f746f6c", "metadata": {}, "source": [ "常熟的斜率为0，所以对常熟求导为0。" ] }, { "cell_type": "markdown", "id": "ae9550c0", "metadata": {}, "source": [ "$$f(x) = 1\\rightarrow \\frac{d}{dx}f(x)=\\frac{d}{dx}1=0$$" ] }, { "cell_type": "code", "execution_count": 19, "id": "ba2fa383", "metadata": {}, "outputs": [], "source": [ "x = symbols('x')" ] }, { "cell_type": "code", "execution_count": 21, "id": "354c80cc", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(1, x)#第一个参数是函数，第二个参数是参数" ] }, { "cell_type": "markdown", "id": "f68ef80d", "metadata": {}, "source": [ "在大学里面，我们用这些符号来表示求导。\n", "$$f'(x)=\\frac{d}{dx}f(x)=\\frac{df}{dx}(x)=\\frac{df(x)}{dx}$$" ] }, { "cell_type": "markdown", "id": "40f31692", "metadata": {}, "source": [ "还记得吗？" ] }, { "cell_type": "markdown", "id": "fa9eb8ad", "metadata": {}, "source": [ "$$f(x) = x\\rightarrow \\frac{d}{dx}f(x)=\\frac{d}{dx}x=\\frac{d}{dx}x^1=1\\cdot x^{1-1}=1\\cdot x^0=1\\cdot 1 = 1$$" ] }, { "cell_type": "code", "execution_count": 22, "id": "fc5a229e", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 1$" ], "text/plain": [ "1" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(x, x)" ] }, { "cell_type": "markdown", "id": "e48e63b5", "metadata": {}, "source": [ "$$f(x) = 2x\\rightarrow \\frac{d}{dx}f(x)=\\frac{d}{dx}2x=2\\cdot\\frac{d}{dx} x=2\\cdot x^{1-1}=2\\cdot x^0=2\\cdot 1 = 2$$" ] }, { "cell_type": "code", "execution_count": 23, "id": "e6e4d2dc", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 2$" ], "text/plain": [ "2" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(2*x, x)" ] }, { "cell_type": "markdown", "id": "85f24aa2", "metadata": {}, "source": [ "上面两个函数的求导结果，我们可以推断。\n", "$$f(x)=mx\\rightarrow f'(x) = m$$" ] }, { "cell_type": "markdown", "id": "1cdd540f", "metadata": {}, "source": [ "$$f(x) = 3x^2\\rightarrow \\frac{d}{dx}f(x)=3\\cdot \\frac{d}{dx}x^2=3\\cdot 2x^{2-1}=3\\cdot2x^1=6x$$" ] }, { "cell_type": "code", "execution_count": 25, "id": "550c8bf2", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 6 x$" ], "text/plain": [ "6*x" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(3*x**2, x)" ] }, { "cell_type": "markdown", "id": "0b47659b", "metadata": {}, "source": [ "$$f(x) = 3x^2 + 5x\\rightarrow \\frac{d}{dx}f(x)=\\\\ \n", "\\frac{d}{dx}[3x^2+5x]= \\\\ \n", "\\frac{d}{dx}3x^2+\\frac{d}{dx}5x^1= \\\\\n", "3\\cdot\\frac{d}{dx}x^2+5\\cdot\\frac{d}{dx}x^1=\\\\\n", "3\\cdot2x^{2-1}+5\\cdot 1x^{1-1}=\\\\\n", "3\\cdot 2x^1 + 5\\cdot x^0 = \\\\\n", "6x + 5\n", "$$" ] }, { "cell_type": "code", "execution_count": 28, "id": "181c7f01", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 6 x + 5$" ], "text/plain": [ "6*x + 5" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(3*x**2+5*x, x)" ] }, { "cell_type": "markdown", "id": "4141e744", "metadata": {}, "source": [ "由上面可以推导。\n", "$$\\frac{d}{dx}[f(x)+g(x)]=\\frac{d}{dx}f(x)+\\frac{d}{dx}g(x)=f'(x)+g'(x)$$" ] }, { "cell_type": "markdown", "id": "722cfa4e", "metadata": {}, "source": [ "求以下函数的导数。\n", "$f(x)=5x^5+4x^3-5\\\\\n", "f(x)=x^3+2x^2-5x-7\n", "$" ] }, { "cell_type": "markdown", "id": "fde6d1b9", "metadata": {}, "source": [ "sigmoid导数如下。" ] }, { "cell_type": "code", "execution_count": 31, "id": "1319f5dc", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\frac{e^{- x}}{\\left(1 + e^{- x}\\right)^{2}}$" ], "text/plain": [ "exp(-x)/(1 + exp(-x))**2" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(1/(1+exp(-x)),x)" ] }, { "cell_type": "markdown", "id": "4863b184", "metadata": {}, "source": [ "relu导数如下。" ] }, { "cell_type": "code", "execution_count": 32, "id": "76afca47", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 1$" ], "text/plain": [ "1" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(x, x)#if x > 0" ] }, { "cell_type": "code", "execution_count": 33, "id": "95d918af", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(0, x)#if x <= 0" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.7.9" } }, "nbformat": 4, "nbformat_minor": 5 }