Let's differentiate the function $f(x) = \sin^{-1}(x) + \ln(\sqrt{1 - x^2})$.

Step-by-step differentiation:

1. Differentiate $\sin^{-1}(x)$: $\frac{d}{dx} \left( \sin^{-1}(x) \right) = \frac{1}{\sqrt{1 - x^2}}$

2. Simplify $\ln(\sqrt{1 - x^2})$:

   $\ln(\sqrt{1 - x^2}) = \frac{1}{2} \ln(1 - x^2)$

   Now, differentiate $\frac{1}{2} \ln(1 - x^2)$ using the chain rule: $\frac{d}{dx} \left( \frac{1}{2} \ln(1 - x^2) \right) = \frac{1}{2} \cdot \frac{1}{1 - x^2} \cdot (-2x) = \frac{-x}{1 - x^2}$

Final derivative:

Combining both parts, the derivative of $f(x)$ is: $f'(x) = \frac{1}{\sqrt{1 - x^2}} - \frac{x}{1 - x^2}$

This is the derivative of the given function.

Do you have any questions or would you like more details?

Here are 5 questions to explore further:

1. How would you differentiate other inverse trigonometric functions?
2. What are the conditions for the derivative of $\sin^{-1}(x)$ to exist?
3. How does the chain rule apply to logarithmic differentiation?
4. How do you differentiate products and quotients of functions?
5. What happens if you differentiate an inverse hyperbolic function?

Tip: When differentiating logarithmic functions, always simplify them as much as possible first for easier application of rules.