Product rule: formula, calculator and worked example

What the symbols mean

$D$: the value of the derivative $(uv)′$ at the chosen x
$m$: in the calculator, the power in $u = x^m$
$k$: in the calculator, the rate in $v = e^(kx)$
$x$: the point where the derivative is evaluated
$u$: the first factor
$v$: the second factor
$u'$: the derivative of u

When to use it

Use it whenever two functions of x are multiplied: $x²e^x$ , $x \sin x$ , $(x + 1)ln x$ . If one factor is a constant you don't need it (the constant just comes along). The calculator handles $x^me^(kx)$ , a common exam family, and shows u, u′, v and v′ separately.

Worked example

Find the slope of f(x) = x²eˣ at x = 1. Here u = xᵐ with m = 2 and v = eᵏˣ with k = 1.

Product rule: differentiate each factor, multiply by the other, and add.
$(u v)' = u' v + u v'$
Here $u = x^{2}$ and $v = e^{x}$ , so $u' = 2 x$ and $v' = e^{x}$ .
Evaluate at x:
$u = x^{m}$
$= 1^{2}$
$= 1$
$du = m x^{m - 1}$
$= 2 \cdot 1^{2 - 1}$
$= 2 \cdot 1^{1}$
$= 2 \cdot 1$
$= 2$
$v = e^{k x}$
$= e^{1 \cdot 1}$
$= e^{1}$
$\approx 2.71828$
$dv = k e^{k x}$
$= 1 e^{1 \cdot 1}$
$= 1 e^{1}$
$\approx 2.71828$
Combine:
$D = u' v + u v'$
$\approx 2 \cdot 2.71828 + 1 \cdot 2.71828$
$\approx 8.15485$

Answer f′(1) = 3e ≈ 8.1548

Why it works

Think of a rectangle with sides u and v. If both sides grow a little, the area grows by $v Δ u$ plus $u Δ v$ (plus a tiny corner $Δ u Δ v$ that vanishes in the limit).

Common mistakes

Multiplying the derivatives: $(uv)′ ≠ u′v′$ . For $x \cdot x$ that would give 1 instead of $2x$ .
Dropping one of the two terms.
Using it when one factor is a constant; $(5x³)′ = 15x²$ directly.

Product rule

Calculator

See it move

What the symbols mean

When to use it

Worked example

Why it works

Common mistakes