Product Rule
The product rule gives the derivative of a product of two functions: (fg)' = f'g + fg'.
Formula
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Definition
When two functions are multiplied together, the product rule says: differentiate the first times keep the second, plus keep the first times differentiate the second. If $f$ and $g$ are differentiable, $(fg)' = f'g + fg'$, proven from the definition by adding and subtracting $f(x+h)g(x)$ inside the limit of the difference quotient. This is the Leibniz rule for derivatives: in the context of differential operators, it defines a derivation on an algebra, $D(fg) = D(f)g + fD(g)$, and generalizes to higher-order derivatives via $(fg)^{(n)} = \sum_{k=0}^{n} C(n,k) f^{(k)} g^{(n-k)}$.
Example
Differentiating $x^2 \sin(x)$ gives $(2x)(\sin x) + (x^2)(\cos x) = 2x\sin x + x^2\cos x$. For $f(x) = e^x x^3$: $f'(x) = e^x x^3 + e^x \cdot 3x^2 = e^x(x^3 + 3x^2)$. The rule extends to three or more factors, $(fgh)' = f'gh + fg'h + fgh'$, and the generalized Leibniz rule gives, for instance, $(x e^x)'' = e^x(x + 2)$.
Key Insight
You cannot just multiply the two derivatives; the product rule accounts for both functions changing at the same time, and the number of terms in the multi-factor version equals the number of factors. The Leibniz rule is preserved under many generalizations including distributions, p-adic derivatives, and the exterior derivative on differential forms.