To find $f'(x)$ for $f(x) = 2^{\sin(x)}$, we will use the chain rule along with the properties of exponential functions.

Step-by-Step Solution:

1. Rewrite $f(x)$ in a form that helps with differentiation: $f(x) = 2^{\sin(x)}$ We can express this using natural logarithms as: $f(x) = e^{\ln(2^{\sin(x)})} = e^{\sin(x) \cdot \ln(2)}$

2. Differentiate using the chain rule: Let $g(x) = \sin(x) \cdot \ln(2)$, so $f(x) = e^{g(x)}$.

   Then, using the chain rule: $f'(x) = e^{g(x)} \cdot g'(x)$

3. Substitute back for $e^{g(x)}$: Since $e^{g(x)} = 2^{\sin(x)}$, we have: $f'(x) = 2^{\sin(x)} \cdot g'(x)$

4. Find $g'(x)$: $g(x) = \sin(x) \cdot \ln(2)$ Differentiating $g(x)$ with respect to $x$: $g'(x) = \cos(x) \cdot \ln(2)$

5. Combine the results: $f'(x) = 2^{\sin(x)} \cdot \cos(x) \cdot \ln(2)$

Final Answer:

The correct choice is: $f'(x) = \ln(2) \cdot 2^{\sin(x)} \cdot \cos(x)$ which matches the second option in the list.

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Related Questions:

1. What is the derivative of $f(x) = a^{\sin(x)}$ for any base $a$?
2. How does the chain rule apply to functions of the form $e^{g(x)}$?
3. What are some examples of applying logarithmic differentiation?
4. How do you differentiate functions of the form $x^{f(x)}$?
5. How can the derivative rules for exponential functions be extended to complex numbers?

Tip: For derivatives of exponential functions with variable exponents, expressing the function in terms of $e$ using logarithmic properties often simplifies differentiation.