1. **Problem statement:** We want to find the derivative of the product of two functions, say $f(x)$ and $g(x)$. That is, find $\frac{d}{dx}[f(x)g(x)]$. 2. **Formula (Product Rule):** The product rule states: $$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$ This means the derivative of the product is the derivative of the first times the second plus the first times the derivative of the second. 3. **Explanation:** When differentiating a product, you cannot just multiply the derivatives. Instead, you apply the product rule to account for how both functions change. 4. **Example:** Suppose $f(x) = x^2$ and $g(x) = \sin x$. 5. Compute derivatives: $$f'(x) = 2x$$ $$g'(x) = \cos x$$ 6. Apply product rule: $$\frac{d}{dx}[x^2 \sin x] = (2x)(\sin x) + (x^2)(\cos x)$$ 7. **Final answer:** $$\boxed{2x \sin x + x^2 \cos x}$$ This is the derivative of the product $x^2 \sin x$ using the product rule.