Product rule derivative: the « u'v + uv' » mnemonic

The formula

If $f(x) = u(x) \cdot v(x)$, then:

$f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$

Short version: $(uv)' = u'v + uv'$.

The mnemonic that sticks

« The derivative of the first, times the second, plus the first times the derivative of the second. »

Chant it three times — you’ll never forget it.

A concrete example

Let’s differentiate $f(x) = (2x + 1)(x^2 - 3)$.

• $u = 2x + 1 \Rightarrow u' = 2$
• $v = x^2 - 3 \Rightarrow v' = 2x$

So:

$f'(x) = u'v + uv' = 2 \cdot (x^2 - 3) + (2x + 1) \cdot 2x$

$= 2x^2 - 6 + 4x^2 + 2x = 6x^2 + 2x - 6$

The classic mistake

Many students write $(uv)' = u' \cdot v'$. Wrong! Check on a simple example:

• $(x \cdot x)' = (x^2)' = 2x$
• But $x' \cdot x' = 1 \cdot 1 = 1$

Doesn’t match — the true formula gives $1 \cdot x + x \cdot 1 = 2x$ ✓

?Your turn

What is the derivative of f(x) = x · sin(x)?

A.cos(x)B.sin(x) + x·cos(x)C.x·cos(x)D.sin(x) · cos(x)

Check