지수함수와 로그함수의 미분

We use logarithmic differentiation; so we modify our function to consider

$$\begin{aligned} \log f(x) &= \log(x) \cdot \cos^3(x) \\ \text{Differentiating, we find:} \\ \dfrac{f'(x)}{f(x)} &= \frac{\mathrm{d}}{\mathrm{d}x} \left\{ \log(x) \cdot \cos^3(x) \right\} = 3\cos^2(x)\cdot (-\sin(x)) \cdot \log(x) + \frac{\cos^3(x)}{x} \\ \text{Finally, we solve for $f'(x)$:} \\ f'(x) &= f(x) \cdot \left( -3\cos^2(x)\sin(x) \log(x) + \frac{\cos^3(x)}{x} \right) \\ &= x^{\cos^3(x)} \cdot \left( -3\cos^2(x)\sin(x) \log(x) + \frac{\cos^3(x)}{x} \right) \end{aligned}$$

Remark: negative numbers behave in a complicated manner when they are the base of an exponential expression. For example, the expression $(-1)^x$ is defined when $x$ is the reciprocal of an odd number (like $x=\frac{1}{5}$ or $x=\frac{1}{7}$), but not when $x$ is the reciprocal of an even number (like $x=\frac{1}{2}$). Since the domain of $f(x)$ was restricted so that $x$ is always positive, we avoid these complications.