1. **Problem:** Find the derivative of $f(x) = \sin(x^2 + 3)$. 2. **Formula and rules:** Use the chain rule: if $f(x) = \sin(g(x))$, then $f'(x) = \cos(g(x)) \cdot g'(x)$. 3. **Intermediate work:** Here, $g(x) = x^2 + 3$, so $g'(x) = 2x$. 4. **Derivative:** $$f'(x) = \cos(x^2 + 3) \cdot 2x = 2x \cos(x^2 + 3)$$ 5. **Explanation:** We first identify the inner function $x^2 + 3$, then differentiate it to get $2x$. We multiply this by the derivative of the outer function $\sin$ which is $\cos$. **Final answer:** $$\boxed{f'(x) = 2x \cos(x^2 + 3)}$$