1. **State the problem:** We are given the function $f(x) = x^2 \sin x$ and want to understand its behavior. 2. **Formula and rules:** The function is a product of $x^2$ and $\sin x$. To analyze it, we can find its derivative using the product rule: $$\frac{d}{dx}[u \cdot v] = u'v + uv'$$ where $u = x^2$ and $v = \sin x$. 3. **Calculate derivatives:** - $u = x^2 \Rightarrow u' = 2x$ - $v = \sin x \Rightarrow v' = \cos x$ 4. **Apply product rule:** $$f'(x) = u'v + uv' = 2x \sin x + x^2 \cos x$$ 5. **Interpretation:** The derivative $f'(x)$ tells us where the function increases or decreases and helps find extrema. 6. **Summary:** The function is $f(x) = x^2 \sin x$ and its derivative is $$f'(x) = 2x \sin x + x^2 \cos x$$ which can be used for further analysis such as finding critical points or graphing. **Final answer:** $$f'(x) = 2x \sin x + x^2 \cos x$$