1. **State the problem:** Find the differential coefficient (derivative) of the function $y = e^{\sin x}$. 2. **Recall the formula:** The derivative of an exponential function $e^{u(x)}$ with respect to $x$ is given by $$\frac{dy}{dx} = e^{u(x)} \cdot \frac{du}{dx}$$ where $u(x)$ is a differentiable function of $x$. 3. **Identify $u(x)$:** Here, $u(x) = \sin x$. 4. **Differentiate $u(x)$:** The derivative of $\sin x$ is $\cos x$, so $$\frac{du}{dx} = \cos x$$. 5. **Apply the chain rule:** Substitute back into the formula: $$\frac{dy}{dx} = e^{\sin x} \cdot \cos x$$. 6. **Final answer:** The differential coefficient of $e^{\sin x}$ is $$\boxed{e^{\sin x} \cos x}$$. This means the rate of change of $e^{\sin x}$ with respect to $x$ depends on both the exponential of $\sin x$ and the cosine of $x$.