Math · Calculus I · Concept

The product rule for derivatives

The product rule says the derivative of f(x)g(x) is f′(x)g(x) + f(x)g′(x): each factor changes in turn while the other keeps its current value. It is not f′(x)g′(x).

The rule

Differentiate the first factor and keep the second, then keep the first factor and differentiate the second, and add the two terms. The rule holds wherever both factors are differentiable.

\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)

\begin{array}{c} \frac{d}{d x} [f (x) g (x)] \\ = f^{'} (x) g (x) + f (x) g^{'} (x) \end{array}

Why there are two terms

Picture f and g as the sides of a rectangle with area fg. When x changes a little, the width grows by Δf and the height by Δg. The new area is the old rectangle plus two edge strips, g·Δf and f·Δg, and a small corner Δf·Δg.

Δ (f g) = g Δ f + f Δ g + Δ f Δ g

Why does the corner disappear?

Divide the change in area by h and let h → 0. The strips become g·f′ and f·g′. The corner becomes (Δf/h)·Δg, which tends to f′(x)·0 = 0 because g is continuous, so it contributes nothing to the derivative.

Multiplying the derivatives does not work

Test f′g′ on x·x = x²: it gives 1·1 = 1, but the derivative of x² is 2x. The product rule gives 1·x + x·1 = 2x, as it should.

\frac{d}{d x} (x \cdot x) = 1 \cdot x + x \cdot 1 = 2 x

Products of different kinds of functions

The product rule is essential when the factors cannot simply be multiplied out, such as x² sin x or eˣ ln x. Combine it with the derivative of each factor.

\frac{d}{d x} (x^{2} sin x) = 2 x sin x + x^{2} cos x

\begin{array}{c} \frac{d}{d x} (x^{2} sin x) \\ = 2 x sin x + x^{2} cos x \end{array}

Three factors

For three factors, differentiate one at a time and add the three terms.

{(f g h)}^{'} = f^{'} g h + f g^{'} h + f g h^{'}

Common mistakes

Writing (fg)′ = f′g′.
Differentiating both factors in the same term, as in f′g′ + fg.
Forgetting the chain rule inside a factor: in x²e^(3x), the derivative of the second factor is 3e^(3x).

Key terms

Product rule: To differentiate a product, take the derivative of the first factor times the second, plus the first times the derivative of the second: (uv)′ = u′v + uv′.
Derivative: The instantaneous rate of change of a function: the limit of the average rate of change as the step shrinks to zero, when that limit exists. On a graph it is the slope of the tangent line.
Differentiable: Having a derivative at a point. A function is continuous wherever it is differentiable, but not the other way round: |x| is continuous at x = 0 with no derivative there.
Chain rule: The rule for differentiating a function inside another function: differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside.

Work through an example

Differentiate f(x) = x²(x + 1) with the product rule, then check the result by expanding first.

Differentiate x²(x + 1) with the product rule →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

OpenStax Calculus Volume 1 — Differentiation rules

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