역삼각함수의 미분

We use logarithmic differentiation, our standard method of differentiating an expression of the form $(\text{function})^{\text{function}}$.

$$\begin{aligned} f(x)&=[\sin x +2]^{\mathop{\mathrm{arcsec}} x}\\ \log(f(x))&=\mathop{\mathrm{arcsec}} x \cdot \log[\sin x +2]\\ \frac{f'(x)}{f(x)}&=\frac{1}{|x|\sqrt{x^2-1}}\log[\sin x +2]+\mathop{\mathrm{arcsec}} x \cdot \frac{\cos x}{\sin x +2}\\ f'(x)&=[\sin x +2]^{\mathop{\mathrm{arcsec}} x}\left(\frac{\log[\sin x +2]}{|x|\sqrt{x^2-1}}+ \frac{\mathop{\mathrm{arcsec}} x \cdot\cos x}{\sin x +2}\right) \end{aligned}$$

The domain of $\mathop{\mathrm{arcsec}} x$ is $|x| \geq 1$. For any $x$, $\sin x +2$ is positive, and a positive number can be raised to any power. (Recall negative numbers cannot be raised to any power–for example, $(-1)^{1/2}=\sqrt{-1}$ is not a real number.) So, the domain of $f(x)$ is $|x| \geq 1$.