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.

Idea clave

$$\big(f g\big)' = f' g + f g'.$$ 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 $\Delta f\,\Delta g$.

Por qué funciona

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.

Ejemplo resuelto: 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.

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

Ejemplo resuelto: 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: $\frac{x^2 + 3 + 4x^2}{2\sqrt{x}} = \frac{5x^2 + 3}{2\sqrt{x}}$.

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

Ejemplo resuelto: From a table

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

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

Respuesta: $7$

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 $\sqrt{x}\,e^x$.

Error común

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.

More factors, and when to expand

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