The Product Rule

The Constant, Power and Sum Rules handles sums easily, because derivatives split over sums. Products do not split so simply. The right rule has two terms, and a picture shows why.

Key idea

$$\big(f g\big)' = f' g + f g'.$$ Differentiate one factor at a time, keep the other as it is, and add the two results.

Area $= f \cdot g$. When $x$ grows, the area gains a strip $g\,\Delta f$, a strip $f\,\Delta g$, and a tiny corner $\Delta f\,\Delta g$.

Why it works

Think of $f(x)g(x)$ as the area of a rectangle with sides $f$ and $g$. When $x$ changes by $\Delta x$, the sides change by $\Delta f$ and $\Delta g$, and the area changes by $$\Delta(fg) = g\,\Delta f + f\,\Delta g + \Delta f\,\Delta g.$$ Divide by $\Delta x$: $g\frac{\Delta f}{\Delta x} + f\frac{\Delta g}{\Delta x} + \Delta f\frac{\Delta g}{\Delta x}$. As $\Delta x \to 0$ the first two terms become $g f' + f g'$, while the last term has an extra factor $\Delta f \to 0$, so it disappears.

Worked example: Two polynomials

Differentiate $(x^2 + 1)(x^3 - 4x)$.

$f = x^2 + 1$, $f' = 2x$; $g = x^3 - 4x$, $g' = 3x^2 - 4$.
$$f'g + fg' = 2x(x^3 - 4x) + (x^2 + 1)(3x^2 - 4)$$.
Expand: $$2x^4 - 8x^2 + 3x^4 - 4x^2 + 3x^2 - 4 = 5x^4 - 9x^2 - 4$$.
Check by expanding first: $x^5 - 3x^3 - 4x$ has derivative $5x^4 - 9x^2 - 4$. They agree.

Answer: $5x^4 - 9x^2 - 4$

Worked example: A root times a polynomial

Differentiate $\sqrt{x}\,(x^2 + 3)$.

$f = x^{1/2}$, $f' = \frac{1}{2\sqrt{x}}$; $g = x^2 + 3$, $g' = 2x$.
$\frac{x^2 + 3}{2\sqrt{x}} + 2x\sqrt{x}$.
Common denominator: $\frac{x^2 + 3 + 4x^2}{2\sqrt{x}} = \frac{5x^2 + 3}{2\sqrt{x}}$.

Answer: $\frac{5x^2 + 3}{2\sqrt{x}}$

Worked example: From a table

$f(2) = 3$, $f'(2) = -1$, $g(2) = 5$, $g'(2) = 4$. Find $(fg)'(2)$.

$$(fg)'(2) = f'(2)g(2) + f(2)g'(2) = (-1)(5) + (3)(4)$$.
$= -5 + 12 = 7$.

Answer: $7$

More factors, and when to expand

With three factors, differentiate one at a time and keep the other two: $(fgh)' = f'gh + fg'h + fgh'$. The area picture becomes a box whose volume grows on three faces. When both factors are short polynomials, multiplying out first and using the power rule is just as good; the product rule shines when a factor cannot be multiplied out, such as $x^2\sin x$ or $\sqrt{x}\,e^x$.

Common mistake

Writing $(fg)' = f'g'$. Test it on $x \cdot x = x^2$: the true derivative is $2x$, but $f'g' = 1 \cdot 1 = 1$. The picture explains the error: the area grows along two sides, so there must be two terms. Check any product rule answer by expanding first, when that is possible.

The Product Rule

More factors, and when to expand

Practice

Builds on

Leads to

Sources and further reading