지수함수와 로그함수의 미분

This requires us to apply the chain rule twice.

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x} \left\{ \sqrt{-\log(\cos x)} \right\} &= \frac{1}{2\sqrt{ -\log(\cos x)}} \cdot \frac{\mathrm{d}}{\mathrm{d}x} \left\{ -\log\left(\cos x\right) \right\}\\ &= -\frac{1}{2\sqrt{-\log(\cos x)}} \cdot \frac{1}{ \cos x} \frac{\mathrm{d}}{\mathrm{d}x} \left\{ \cos x \right\}\\ &= -\frac{1}{2\sqrt{-\log(\cos x)}} \cdot \frac{1}{\cos x} \cdot \left(-\sin x\right)\\ &=\frac{\tan x}{2\sqrt{-\log(\cos x)}} \end{aligned}$$

Remark: it looks strange to see a negative sign in the argument of a square root. Since the cosine function always gives values that are at most 1, $\log(\cos x)$ is always negative or zero over its domain. So, $\sqrt{\log(\cos x)}$ is only defined for the points where $\cos x=1$ (and so $\log(\cos x) = 0$–this isn't a very interesting function! In contrast, $-\log(\cos x)$ is always positive or zero over its domain – and therefore we can always take its square root.