单变量函数的微分

f'(x) = \frac{df(x)}{dx} = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}

f'(x) = \frac{f(x + dx) - f(x - dx)}{2 dx}

v(x) = \frac{df(x)}{dx}

1import numpy as np
2import matplotlib.pyplot as plt
3from scipy.misc import derivative
4
5# 目标函数
6def objective(x):
7    return (x + 2) * (x - 6)
8
9x0 = 5  # 在该点计算导数
10y0 = objective(x0)
11dfy = derivative(objective, x0, dx=1e-6)  # 计算导数
12
13print('dy/dx 在 x = %.3f 处的值为 %.3f' % (x0, dfy))
14
15# 绘制曲线以进行图形化可视化
16plt.figure(figsize=(10, 10))
17plt.rcParams.update({'font.size': 22})
18xs = np.arange(-8, 12, 0.1)
19plt.plot(xs, objective(xs), linewidth=2, label='y = (x+2)*(x-6)')
20
21# 绘制在点 x0 处与曲线相切且斜率为 dy/dx 的切线
22tangent_fn = lambda x: dfy * x + (y0 - dfy * x0)
23xs = np.arange(-3, 11, 0.1)
24plt.plot(xs, tangent_fn(xs), linewidth=2, ls='-', color='red', label='dy/dx')
25
26# 绘制点 (x0, y0)，以及 x 轴和 y 轴
27plt.scatter(x0, y0, label='(x,y) = ({},{})'.format(x0, y0))
28plt.axhline(y=0, ls='-', color='black')
29plt.axvline(x=0, ls='-', color='black')
30plt.legend()
31plt.savefig('./output/plot.png')