1. **Problem:** Find the derivative of $y = e^{\sin 4x}$. 2. **Formula and rules:** The derivative of $e^u$ with respect to $x$ is $e^u \cdot \frac{du}{dx}$ (chain rule). 3. **Step-by-step solution:** - Let $u = \sin 4x$. - Then $\frac{du}{dx} = \cos 4x \cdot 4 = 4 \cos 4x$. - Therefore, $\frac{dy}{dx} = e^{\sin 4x} \cdot 4 \cos 4x$. 4. **Final answer:** $$\frac{dy}{dx} = 4 e^{\sin 4x} \cos 4x$$