# 链式法则

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

$(f \circ g)'(x) = f'(g(x)) g'(x).$

## 例子

 $h '(x) \,$ $= h '(g(x)) g ' (x) \,$ $= 3(g(x))^2(2x) \,$ $= 3(x^2 + 1)^2(2x) \,$ $= 6x(x^2 + 1)^2. \,$

$\frac{d}{dx}\arctan\,x\,=\,\frac{1}{1+x^2}$
$\frac{d}{dx}\arctan\,f(x)\,=\,\frac{f'(x)}{1+f^2(x)}$
$\frac{d}{dx}\arctan\,\sin\,x\,=\,\frac{\cos\,x}{1+\sin^2\,x}$

## 证明

fg为函数，x为常数，使得fg(x)可导，且gx可导。根据可导的定义，

$g(x+\delta)-g(x)= \delta g'(x) + \epsilon(\delta)\delta \,$，其中当$\delta\to 0$时，$\epsilon(\delta) \to 0 \,$

$f(g(x)+\alpha) - f(g(x)) = \alpha f'(g(x)) + \eta(\alpha)\alpha \,$，其中当$\alpha\to 0. \,$时，$\eta(\alpha) \to 0 \,$

 $f(g(x+\delta))-f(g(x))\,$ $= f(g(x) + \delta g'(x)+\epsilon(\delta)\delta) - f(g(x)) \,$ $= \alpha_\delta f'(g(x)) + \eta(\alpha_\delta)\alpha_\delta \,$

$\frac{f(g(x+\delta))-f(g(x))}{\delta} \to g'(x)f'(g(x)).$

## 多元复合函数求导法则

${\ dz \over dt}={\partial z \over \partial x}{dx \over dt}+{\partial z \over \partial y}{dy \over dt}.$

${\partial z \over \partial x}={\partial z \over \partial u}{\partial u \over \partial x}+{\partial z \over \partial v}{\partial v \over \partial x}$
${\partial z \over \partial y}={\partial z \over \partial u}{\partial u \over \partial y}+{\partial z \over \partial v}{\partial v \over \partial y}.$

$\vec r = (u,v)$

$\frac{\partial f}{\partial x}=\vec \nabla f \cdot \frac{\partial \vec r}{\partial x}.$

$\frac{\partial(z_1,\ldots,z_m)}{\partial(x_1,\ldots,x_p)} = \frac{\partial(z_1,\ldots,z_m)}{\partial(y_1,\ldots,y_n)} \frac{\partial(y_1,\ldots,y_n)}{\partial(x_1,\ldots,x_p)}.$

## 高阶导数

$\frac{d (f \circ g) }{dx} = \frac{df}{dg}\frac{dg}{dx}$
$\frac{d^2 (f \circ g) }{d x^2} = \frac{d^2 f}{d g^2}\left(\frac{dg}{dx}\right)^2 + \frac{df}{dg}\frac{d^2 g}{dx^2}$
$\frac{d^3 (f \circ g) }{d x^3} = \frac{d^3 f}{d g^3} \left(\frac{dg}{dx}\right)^3 + 3 \frac{d^2 f}{d g^2} \frac{dg}{dx} \frac{d^2 g}{d x^2} + \frac{df}{dg} \frac{d^3 g}{d x^3}$
$\frac{d^4 (f \circ g) }{d x^4} =\frac{d^4 f}{dg^4} \left(\frac{dg}{dx}\right)^4 + 6 \frac{d^3 f}{d g^3} \left(\frac{dg}{dx}\right)^2 \frac{d^2 g}{d x^2} + \frac{d^2 f}{d g^2} \left\{ 4 \frac{dg}{dx} \frac{d^3 g}{dx^3} + 3\left(\frac{d^2 g}{dx^2}\right)^2\right\} + \frac{df}{dg}\frac{d^4 g}{dx^4}.$