Let's solve this problem by differentiating the function $f(x) = \sin(5 \ln(x))$ with respect to $x$.

Solution

1. Identify the outer and inner functions:

   • The outer function is $\sin(u)$ where $u = 5 \ln(x)$.
   • The inner function is $u = 5 \ln(x)$.

2. Apply the chain rule: According to the chain rule: $f'(x) = \cos(5 \ln(x)) \cdot \frac{d}{dx}[5 \ln(x)]$

3. Differentiate the inner function $5 \ln(x)$: $\frac{d}{dx}[5 \ln(x)] = 5 \cdot \frac{1}{x} = \frac{5}{x}$

4. Combine the results: Substitute back into the derivative: $f'(x) = \cos(5 \ln(x)) \cdot \frac{5}{x}$

5. Final Answer: $f'(x) = \frac{5 \cos(5 \ln(x))}{x}$

So, the derivative of $f(x) = \sin(5 \ln(x))$ is: $f'(x) = \frac{5 \cos(5 \ln(x))}{x}$

Would you like further details on the steps, or do you have any additional questions?

Here are some related questions to practice:

1. Differentiate $g(x) = \cos(3 \ln(x))$.
2. Differentiate $h(x) = e^{2 \ln(x)}$.
3. Find the second derivative of $f(x) = \sin(5 \ln(x))$.
4. Differentiate $k(x) = \tan(4 \ln(x))$.
5. Solve $f'(x) = 0$ for $f(x) = \sin(5 \ln(x))$.

Tip: For functions involving $\ln(x)$ inside trigonometric or exponential functions, the chain rule is essential to handle the differentiation.