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# 求函数微分的解题策略

$\frac{d}{dx}\left[{x}^{n}\right]=n\cdot {x}^{n-1}$
$\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]={f}^{\prime }\left(x\right)+{g}^{\prime }\left(x\right)$
$\frac{d}{dx}\left[f\left(x\right)\cdot g\left(x\right)\right]={f}^{\prime }\left(x\right)g\left(x\right)+f\left(x\right){g}^{\prime }\left(x\right)$
$\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{{f}^{\prime }\left(x\right)g\left(x\right)-f\left(x\right){g}^{\prime }\left(x\right)}{\left[g\left(x\right){\right]}^{2}}$
$\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]={f}^{\prime }\left(g\left(x\right)\right)\cdot {g}^{\prime }\left(x\right)$

## 找出积、商及复合函数

$\begin{array}{rl}& \phantom{=}\frac{d}{dx}\left[\left({x}^{2}+5x\right)\cdot \mathrm{sin}\left(x\right)\right]\\ \\ & =\frac{d}{dx}\left[{x}^{2}+5x\right]\cdot \frac{d}{dx}\left[\mathrm{sin}\left(x\right)\right]\\ \\ & =\left(2x+5\right)\cdot \mathrm{cos}\left(x\right)\\ \\ & =2x\cdot \mathrm{cos}\left(x\right)+5\cdot \mathrm{cos}\left(x\right)\end{array}$

### 常见错误：忘记应用乘法或除法法则。

$\begin{array}{rl}& \phantom{=}\frac{d}{dx}\left[\mathrm{sin}\left({x}^{2}+5x\right)\right]\\ \\ & =\frac{d}{dx}\left[\mathrm{sin}\left(x\right)\cdot \left({x}^{2}+5x\right)\right]\\ \\ & =\frac{d}{dx}\left[\mathrm{sin}\left(x\right)\right]\cdot \left({x}^{2}+5x\right)+\mathrm{sin}\left(x\right)\cdot \frac{d}{dx}\left[{x}^{2}+5x\right]\\ \\ & =\mathrm{cos}\left(x\right)\left({x}^{2}+5x\right)+\mathrm{sin}\left(x\right)\left(2x+5\right)\end{array}$

## 我们可以改写函数来简化微分。

### 有时我们可以将积改写为简单的多项式。

$\begin{array}{rl}& \phantom{=}\frac{d}{dx}\left[\left(x+5\right)\left(x-3\right)\right]\\ \\ & =\frac{d}{dx}\left[x+5\right]\cdot \left(x-3\right)+\left(x+5\right)\cdot \frac{d}{dx}\left[x-3\right]\\ \\ & =\left(1\right)\left(x-3\right)+\left(x+5\right)\left(1\right)\\ \\ & =x-3+x+5\\ \\ & =2x+2\end{array}$$\begin{array}{rl}& \phantom{=}\frac{d}{dx}\left[\left(x+5\right)\left(x-3\right)\right]\\ \\ & =\frac{d}{dx}\left[{x}^{2}+2x-15\right]\\ \\ & =2x+2\end{array}$

$f\left(x\right)=\left(3-8x\right)\left(2x-7\right)$

### 同理，也可以改写除法法则的问题来应用幂法则。

$f\left(x\right)=\frac{{x}^{5}-2{x}^{3}-8{x}^{2}}{x}$

### 最后的例子： 改写商为积

$\begin{array}{rl}\frac{\sqrt{x+3}}{{x}^{4}}& =\sqrt{x+3}\cdot \frac{1}{{x}^{4}}\\ \\ & =\sqrt{x+3}\cdot {x}^{-4}\end{array}$

$h\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{3x}$