Topic 2.8 The Product Rule: differentiate a product of two functions using the product rule.

What this topic is asking

The College Board (Topic 2.8) introduces the product rule for differentiating a product of two functions. The key insight is that the derivative of a product is not the product of the derivatives; you need a specific formula that keeps both factors involved.

The rule

The two-term structure is essential. Each term differentiates exactly one factor while leaving the other unchanged, then the terms are added.

Why not just $u'v'$

A reliable procedure

Differentiating a product cleanly is a matter of bookkeeping: write down $u$ and $v$, compute $u'$ and $v'$ separately off to the side, then plug into $u'v + uv'$ and simplify. Labelling the pieces prevents the common mistake of differentiating both factors at once.

When you need it

Use the product rule whenever two non-constant functions are multiplied: power times trig, power times exponential, trig times exponential, and so on. If one factor is a constant, you do not need the product rule - the constant-multiple rule suffices. For three factors, apply the product rule in stages, treating two of them as a single grouped factor.

A closer look at why the formula has two terms

The two-term shape of the product rule is not a quirk of notation; it reflects how a product responds to a small change in the input. If $x$ nudges by a tiny amount, the value $u(x)$ shifts a little and the value $v(x)$ shifts a little, and the product $uv$ changes for both reasons at once. One contribution comes from $u$ changing while $v$ holds roughly steady, which produces the $u'v$ term; the other comes from $v$ changing while $u$ holds roughly steady, which produces the $uv'$ term. The tiny cross-term where both change simultaneously vanishes in the limit, leaving exactly $u'v + uv'$. Seeing the rule this way explains why you can never get the right derivative by multiplying $u'$ and $v'$ alone: that would account for neither factor holding still, which is not how a small change actually distributes across a product. It is the same reason the area of a growing rectangle increases by "length times change in width plus width times change in length", a picture worth keeping in mind.

Order does not matter, but completeness does

Because addition is commutative, $u'v + uv'$ and $uv' + u'v$ are the same, so it does not matter which factor you call $u$ and which you call $v$. What matters is that both terms appear and each differentiates exactly one factor. A reliable habit on the exam is to write the four pieces $u$, $u'$, $v$, $v'$ in a small table before assembling, so you never accidentally differentiate both factors in the same term or forget a term entirely. When the factors themselves are sums (for example $(3x^2 + 1)\sin x$), keep them grouped in parentheses through the substitution and only expand at the end, which avoids dropping signs during simplification.