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# 链式法则

$\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]={f}^{\prime }\left(g\left(x\right)\right){g}^{\prime }\left(x\right)$

## 快速回顾什么是复合函数

$g$$f$里的函数，所以我们称$g$为“内” 函数，$f$为“外”函数。

$g\left(x\right)=\mathrm{ln}\left(\mathrm{sin}\left(x\right)\right)$是一个复合函数吗？如果是，“内”函数和“外”函数分别是什么？

### 常见错误：无法识别出复合函数

$h\left(x\right)={\mathrm{cos}}^{2}\left(x\right)$是一个复合函数吗？如果是，“内”函数和“外”函数分别是什么？

## 链式法则使用实例

$\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]={f}^{\prime }\left(g\left(x\right)\right)\cdot {g}^{\prime }\left(x\right)$

$\begin{array}{rl}{g}^{\prime }\left(x\right)& =-6\\ \\ {f}^{\prime }\left(x\right)& =5{x}^{4}\end{array}$

$\begin{array}{rl}& \frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]\\ \\ =& {f}^{\prime }\left(g\left(x\right)\right)\cdot {g}^{\prime }\left(x\right)\\ \\ =& 5\left(5-6x{\right)}^{4}\cdot -6\\ \\ =& -30\left(5-6x{\right)}^{4}\end{array}$

### 练习使用链式法则

$\mathrm{sin}\left(2{x}^{3}-4x\right)$的内外函数是什么？

$\frac{d}{dx}\left[\sqrt{\mathrm{cos}\left(x\right)}\right]=\phantom{\rule{0.167em}{0ex}}?$

$x$$f\left(x\right)$$h\left(x\right)$${f}^{\prime }\left(x\right)$${h}^{\prime }\left(x\right)$
$-1$$9$$-1$$-5$$-6$
$2$$3$$-1$$1$$6$
$G\left(x\right)=f\left(h\left(x\right)\right)$
${G}^{\prime }\left(2\right)=$

$\frac{d}{dx}\left[\left(2{x}^{2}-4{\right)}^{3}\right]=3\left(2{x}^{2}-4{\right)}^{2}$