1. We need to find the derivative of $y = e^{\sin x}$ with respect to $z = \cos x$ at $x = \frac{\pi}{2}$. 2. Using the chain rule, $$\frac{dy}{dz} = \frac{dy/dx}{dz/dx}.$$ 3. First, compute $\frac{dy}{dx}$: $$\frac{dy}{dx} = e^{\sin x} \cdot \cos x$$ 4. Next, compute $\frac{dz}{dx}$: $$\frac{dz}{dx} = -\sin x$$ 5. Therefore, $$\frac{dy}{dz} = \frac{e^{\sin x} \cos x}{-\sin x} = -e^{\sin x} \frac{\cos x}{\sin x} = -e^{\sin x} \cot x$$ 6. Evaluate this at $x = \frac{\pi}{2}$: - $\sin \frac{\pi}{2} = 1$ - $\cos \frac{\pi}{2} = 0$ - $\cot \frac{\pi}{2} = \frac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}} = 0$ 7. Substitute these values: $$\frac{dy}{dz} = -e^{1} \cdot 0 = 0$$ 8. The derivative of $e^{\sin x}$ with respect to $\cos x$ at $x=\frac{\pi}{2}$ is $0$. Final answer: **C) 0**