[人工智能] 用python学习微积分（四）链式法则及高阶导数（上）- 通用公式的推导

一、公式推导

1、乘法法则 (uv)' = u'v + v'u

d(uv)'/dx = $\frac{u(x+\Delta x) \times v(x+\Delta x) - u(x)\times v(x)}{\Delta x}$

$=\frac{u(x+\Delta x) \times v(x+\Delta x) - u(x) \times v(x+\Delta x) + u(x) \times v(x+\Delta x) - u(x) \times v(x)}{\Delta x}$

(这里分子中凭空添加2项u(x)v(x+Δx) 一正一负)

$= \frac{v(x+\Delta x) \times (u(x+\Delta x) - u(x)) + u(x) \times (v (x+\Delta x) -v(x))}{\Delta x}$

$= \frac{v(x+\Delta x) \times \Delta u}{\Delta x} + u(x) \times \frac{\Delta v}{\Delta x}$ (由于Δx =>0)

$\doteq v(x+\Delta x)\times u' + u(x)\times v'$ = vu' + u v'

这里讨论下， u' 实际是u(x)' 所以是u(x)的导数，所以vu' + uv'应该分别用python对u(x)和v(x)求导

举个栗子： $u(x) = x^{2} + 5; v(x) =1/ x^3;z(x) = u(x)\times v(x);$

对z(x)求导：根据公式 z(x)' = u(x)'v(x) + v(x)'u(x)，

在python中先后求得u(x)' 和v(x)'的导数

from sympy import *
x= symbols('x')
gfg_exp = x**2+5
dif = diff(gfg_exp, x)
dif

from sympy import *
x= symbols('x')
gfg_exp = 1 / x**3
dif = diff(gfg_exp, x)
dif

将结果带入： z(x)' = u(x)'v(x) + v(x)'u(x) = 2x * v(x) + -3/x**4 * u(x)

x, u, v = symbols('x u v')
expr = 2 * x * v + -3/x**4 * u
expr.subs(v,1/x**3).subs(u, x**2+5)

2、除法法则 $(\frac{u}{v})' =\frac{u'v - v'u}{v^2}$

$\Delta (\frac{u}{v}) = \frac{u+\Delta u}{v+\Delta v} - \frac{u}{v}$

$=\frac{v(u+\Delta u)-u(v+\Delta v)}{v(v+\Delta v)}$

$\frac{\Delta (\frac{u}{v})}{\Delta x} = \frac{\frac{v(u+\Delta u)}{\Delta x}-\frac{u(v+\Delta v)}{\Delta x}}{v(v+\Delta v)}$ 当Δx趋近0

$=\frac{vu' - uv'}{v^2}$

3、除法法则应用

$(\frac{1}{v})' = \frac{1'v - v'1}{v^2} = -v'v^{-2}$

$(\frac{1}{x^n})' = \frac{1' \times x^n - nx^{n-1} \times1}{x^{2n}} = -nx^{-1-n}$

加:2021-12-05 12:02:52 更:2021-12-05 12:04:50

-2024/11/27 3:56:49-

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