1. **State the problem:** Differentiate the function $f(x) = \sin(x^2 + 5)$. 2. **Recall the differentiation rule:** The derivative of $\sin(u)$ with respect to $x$ is $\cos(u) \cdot \frac{du}{dx}$, where $u$ is a function of $x$. 3. **Identify the inner function:** Here, $u = x^2 + 5$. 4. **Compute the derivative of the inner function:** $\frac{du}{dx} = 2x$. 5. **Apply the chain rule:** $$\frac{d}{dx} \sin(x^2 + 5) = \cos(x^2 + 5) \cdot 2x$$ 6. **Final answer:** $$f'(x) = 2x \cos(x^2 + 5)$$