Product Rule
The Formula
If $f$ and $g$ are differentiable, then:
$$\frac{d}{dx}[f(x) \cdot g(x)] = f(x)g'(x) + f'(x)g(x)$$
Or more compactly: $(fg)' = fg' + f'g$
Mnemonic
“First times derivative of second, plus second times derivative of first”
Examples
Example 1
$$\frac{d}{dx}[x^2 \sin x] = x^2 \cos x + 2x \sin x$$
Example 2
$$\frac{d}{dx}[e^x \ln x] = e^x \cdot \frac{1}{x} + e^x \cdot \ln x = e^x\left(\frac{1}{x} + \ln x\right)$$
the derivative of a product is the product of the derivatives.
This is the multiplicative-not-additive error. Derivatives are additive over sums -- $(f+g)' = f' + g'$ is true -- but they are NOT multiplicative over products. $(fg)' = f'g'$ fails because both functions change simultaneously: when $f$ changes, $g$ is still present as a factor. The correct formula picks up both contributions: $(fg)' = fg' + f'g$. The product of $f(x) = e^x$ and $g(x) = x^2$ gives $(e^x x^2)' = e^x \cdot 2x + e^x \cdot x^2 = e^x(x^2 + 2x)$, not $e^x \cdot 2x = 2xe^x$.
the Product Rule and Chain Rule are interchangeable.
This is the composition-is-not-chaining error in reverse. The Product Rule applies when two separate functions are MULTIPLIED: $f(x) \cdot g(x)$. The Chain Rule applies when one function is COMPOSED inside another: $f(g(x))$. For $h(x) = x^2 \sin x$: this is a product, so use the Product Rule: $(x^2)' \sin x + x^2 (\sin x)' = 2x \sin x + x^2 \cos x$. For $h(x) = \sin(x^2)$: this is a composition, so use the Chain Rule: $\cos(x^2) \cdot 2x$. The two rules solve structurally different problems and give different answers when confused.
Why It’s Not $(fg)' = f'g'$
Consider $f(x) = x$ and $g(x) = x$. Then $fg = x^2$.
If the derivative of a product were the product of derivatives: $(fg)' = f'g' = 1 \cdot 1 = 1$ ❌
But we know $(x^2)' = 2x$ ✓
The product rule gives: $fg' + f'g = x(1) + (1)x = 2x$ ✓
Extended Product Rule
For three functions: $$(fgh)' = f'gh + fg'h + fgh'$$