[1],[$f(x)= sqrt(e^x/(ln x))$], [$f'(x)=$], ...

1. $f(x)=\sqrt{\dfrac{e^x}{\ln x}}=\left(\dfrac{e^x}{\ln x}\right)^{1/2}$

$$f'(x)=\frac{e^{x/2}(x\ln x-1)}{2x(\ln x)^{3/2}}$$

2. $f(x)=\dfrac{1}{x\sin x}=(x\sin x)^{-1}$

$$f'(x)=-\frac{\sin x+x\cos x}{x^2\sin^2 x}$$

3. $f(x)=\cos(\sin x)$

$$f'(x)=-\sin(\sin x)\,\cos x$$

4. $f(x)=(2x^2+3)^3$

$$f'(x)=12x(2x^2+3)^2$$

5. $f(x)=e^{1/x}$

$$f'(x)=-\frac{e^{1/x}}{x^2}$$

6. $f(x)=\ln(\sin x+\cos x)$

$$f'(x)=\frac{\cos x-\sin x}{\sin x+\cos x}$$

下面给你 3 组和你刚才那类（链式法则 + 乘除 + 指数/对数/三角复合）难度相近的求导题，每题都写成“$f(x)=$ …，求 $f'(x)$”。

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A组

1. $f(x)=\sqrt{\dfrac{e^{2x}}{\ln x}}$，$f'(x)=\ \ \ ?$
2. $f(x)=\dfrac{1}{x^2\cos x}$，$f'(x)=\ \ \ ?$
3. $f(x)=\sin(\cos x)$，$f'(x)=\ \ \ ?$
4. $f(x)=(3x^2-5)^4$，$f'(x)=\ \ \ ?$
5. $f(x)=e^{-1/x}$，$f'(x)=\ \ \ ?$
6. $f(x)=\ln(2\sin x+\cos x)$，$f'(x)=\ \ \ ?$

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B组

1. $f(x)=\sqrt{\dfrac{e^{x}}{\ln(2x)}}$，$f'(x)=\ \ \ ?$
2. $f(x)=\dfrac{1}{x\sin(2x)}$，$f'(x)=\ \ \ ?$
3. $f(x)=\cos(\tan x)$，$f'(x)=\ \ \ ?$
4. $f(x)=(x^3+1)^5$，$f'(x)=\ \ \ ?$
5. $f(x)=e^{\frac{2}{x}}$，$f'(x)=\ \ \ ?$
6. $f(x)=\ln(\sin x-\cos x)$，$f'(x)=\ \ \ ?$

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C组

1. $f(x)=\sqrt{\dfrac{e^{3x}}{(\ln x)^2}}$，$f'(x)=\ \ \ ?$
2. $f(x)=\dfrac{1}{x^2\sin x}$，$f'(x)=\ \ \ ?$
3. $f(x)=\sin(\sin x)$，$f'(x)=\ \ \ ?$
4. $f(x)=(5-2x^2)^3$，$f'(x)=\ \ \ ?$
5. $f(x)=e^{\frac{\ln x}{x}}$，$f'(x)=\ \ \ ?$
6. $f(x)=\ln(\sin x+\cos 2x)$，$f'(x)=\ \ \ ?$

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如果你愿意，我也可以把其中任意一组的答案（或详细步骤）给你，或者再出更难一点的“多层复合 + 对数求导”版本。

[1] $f(x)=\sqrt{9-25x^2}$

$$f'(x)=\frac{1}{2\sqrt{9-25x^2}}\cdot(-50x)= -\frac{25x}{\sqrt{9-25x^2}}$$

[2] $f(x)=\dfrac{1}{x\sin x}=(x\sin x)^{-1}$

$$f'(x)=-\frac{(x\sin x)'}{(x\sin x)^2}=-\frac{\sin x+x\cos x}{x^2\sin^2 x}$$

[3] $f(x)=\cos(\sin x)$

$$f'(x)=-\sin(\sin x)\cdot(\sin x)'=-\sin(\sin x)\cos x$$

[4] $f(x)=(2x^2+3)^3$

$$f'(x)=3(2x^2+3)^2\cdot(4x)=12x(2x^2+3)^2$$

[5] $f(x)=e^{e^x}$

$$f'(x)=e^{e^x}\cdot(e^x)=e^{e^x+x}$$

（也可写成 $f'(x)=e^{e^x}\,e^x$）

[6] $f(x)=\ln(\sin x+\cos x)$

$$f'(x)=\frac{(\sin x+\cos x)'}{\sin x+\cos x}=\frac{\cos x-\sin x}{\sin x+\cos x}$$