To find the derivative of \( 5 \tan(\ln x) \), we will use the chain rule. Let \( u = \ln x \). Then, the function becomes \( 5 \tan(u) \). First, find the derivative of \( 5 \tan(u) \) with respect to \( u \): \[ \frac{d}{du} [5 \tan(u)] = 5 \sec^2(u) \] Next, find the derivative of \( u = \ln x \) with respect to \( x \): \[ \frac{du}{dx} = \frac{1}{x} \] Now, apply the chain rule: \[ \frac{d}{dx} [5 \tan(\ln x)] = 5 \sec^2(\ln x) \cdot \frac{1}{x} \] So, the derivative is: \[ \frac{5 \sec^2(\ln x)}{x} \] Therefore, the correct answer is: \[ \boxed{C} \]

To find the derivative of $5 \tan(\ln x)$, we will use the chain rule. Let $u = \ln x$. Then, the function becomes $5 \tan(u)$. First, find the derivative of $5 \tan(u)$ with respect to $u$: $\frac{d}{du} [5 \tan(u)] = 5 \sec^2(u)$ Next, find the derivative of $u = \ln x$ with respect to $x$: $\frac{du}{dx} = \frac{1}{x}$ Now, apply the chain rule: $\frac{d}{dx} [5 \tan(\ln x)] = 5 \sec^2(\ln x) \cdot \frac{1}{x}$ So, the derivative is: $\frac{5 \sec^2(\ln x)}{x}$ Therefore, the correct answer is: $\boxed{C}$