1. **Problem 1: Find the derivative $\frac{dy}{dx}$ of $f(x) = e^{\sin(x)}$.** 2. The formula for the derivative of an exponential function with a function in the exponent is: $$\frac{d}{dx} e^{g(x)} = e^{g(x)} \cdot g'(x)$$ 3. Here, $g(x) = \sin(x)$, so we need to find $g'(x)$: $$g'(x) = \cos(x)$$ 4. Applying the chain rule: $$\frac{dy}{dx} = e^{\sin(x)} \cdot \cos(x)$$ 5. Therefore, the correct answer is option c: $e^{\sin(x)} \cos(x)$. 6. **Problem 2: Find the value of $\log_5 25$.** 7. Recall the definition of logarithm: $$\log_a b = c \iff a^c = b$$ 8. We want to find $c$ such that: $$5^c = 25$$ 9. Since $25 = 5^2$, it follows that: $$5^c = 5^2 \implies c = 2$$ 10. Therefore, $\log_5 25 = 2$.