Product Rule
Calculus & Advanced MathThe 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.
Example
Differentiate $x^2 \sin(x)$: $(2x)(\sin x) + (x^2)(\cos x) = 2x\sin x + x^2\cos x$.
Key Insight
You cannot just multiply the two derivatives. The product rule accounts for both functions changing at the same time.
Definition
If $f$ and $g$ are differentiable, then $(fg)' = f'g + fg'$. This is proven from the definition by adding and subtracting $f(x+h)g(x)$ inside the limit of the difference quotient.
Example
$f(x) = e^x x^3$: $f'(x) = e^x x^3 + e^x \cdot 3x^2 = e^x(x^3 + 3x^2)$.
Key Insight
The product rule extends to products of three or more functions: $(fgh)' = f'gh + fg'h + fgh'$. The number of terms equals the number of factors.
Definition
The product rule is the Leibniz rule for derivatives. In the context of differential operators, it defines a derivation on an algebra: $D(fg) = D(f)g + fD(g)$. It generalizes to higher-order derivatives via the generalized Leibniz rule: $(fg)^{(n)} = \sum_{k=0}^{n} C(n,k) f^{(k)} g^{(n-k)}$.
Example
Generalized: $(x e^x)'' = e^x(x + 2)$. Using the Leibniz rule: $C(2,0)(x)(e^x)'' + C(2,1)(1)(e^x)' + C(2,2)(0)(e^x) = xe^x + 2e^x$.
Key Insight
The Leibniz rule is preserved under many generalizations including distributions, p-adic derivatives, and the exterior derivative on differential forms.