The Product Rule: Differentiating Multiplications

Differentiate products without guessing by using a simple two-part rule that matches the graph every time. Build intuition from tiny changes, then apply a repeatable workflow that also extends cleanly to three or more factors.

Differentiating a multiplication is where a lot of instincts break. If you see $f(x)g(x)$, it feels like the derivative should be $f'(x)g'(x)$. That would be convenient, but it misses how slopes behave when two moving quantities are multiplied. Even with something as familiar as $x\cdot x$, the slope you get from the graph does not line up with that naive rule.

To make the mismatch concrete, compare the tangent slopes you would predict for $(x)(x)$ versus the actual slopes you see for $x^2$.

The point is not that $x\cdot x$ is special. It is that multiplication couples the changes of both factors, so the slope has two contributions, not one.

The product rule in plain language

Product rule: if $h(x)=f(x)g(x)$, then

$$(fg)'=f'g+fg'.$$

Read it like this. When you differentiate a product, each factor gets a turn being differentiated while the other factor stays as it is, then you add the results.

There are three common candidates people try in their heads.

• $f'(x)g'(x)$, the tempting but wrong multiply-the-derivatives idea
• $f'(x)g(x)+f(x)g'(x)$, the product rule
• sometimes expanding first, which only works when expansion is actually simpler

Seeing all three side by side helps you build the reflex that the correct derivative has two terms. Use the comparison to check which expression matches real examples.

Two turns
Derivatives do not distribute over multiplication the way they distribute over addition. Products produce a two-term answer because two separate changes matter.

Where the rule comes from

The product rule is not a magical exception. It comes from doing the same thing derivatives always do, measuring how a function changes under a tiny input change.

Let $h(x)=f(x)g(x)$. Increase $x$ by a small amount $\Delta x$. Then both $f$ and $g$ change:

• $f$ becomes $f+\Delta f$
• $g$ becomes $g+\Delta g$

So the product becomes

$$(f+\Delta f)(g+\Delta g)=fg+g\Delta f+f\Delta g+(\Delta f)(\Delta g).$$

Now subtract the original $fg$. The change in the product is

$$\Delta h=g\Delta f+f\Delta g+(\Delta f)(\Delta g).$$

Divide by $\Delta x$ to make it a rate of change:

$$\frac{\Delta h}{\Delta x}=g\frac{\Delta f}{\Delta x}+f\frac{\Delta g}{\Delta x}+\frac{(\Delta f)(\Delta g)}{\Delta x}.$$

As $\Delta x\to0$, the fractions $\frac{\Delta f}{\Delta x}$ and $\frac{\Delta g}{\Delta x}$ approach $f'$ and $g'$. The last term is the key. Since $\Delta f$ and $\Delta g$ both shrink toward $0$, their product shrinks even faster, so $\frac{(\Delta f)(\Delta g)}{\Delta x}$ goes to $0$ under the usual smoothness assumptions you have in basic calculus. That is why the cross-term disappears in the limit, but the two main terms remain.

If you want to see that expansion and which pieces survive as $\Delta x$ gets tiny, step through it interactively.

How to apply it reliably

The hardest part is rarely the algebra. It is choosing $f$ and $g$ cleanly, then staying consistent.

A repeatable workflow

Pick $f(x)$ and $g(x)$ so your expression is literally $f\cdot g$.

• Write $f(x)$ and $g(x)$ on separate lines
• Differentiate each. Keep them aligned as $f'(x)$ and $g'(x)$
• Plug into $f'g+fg'$
• Simplify only after you have both terms written

One example all the way through

Differentiate $h(x)=(x^2+1)\sin x$.

Let $f(x)=x^2+1$ and $g(x)=\sin x$. Then $f'(x)=2x$ and $g'(x)=\cos x$.

So

$$h'(x)=f'g+fg'=2x\sin x+(x^2+1)\cos x.$$

Notice what you did not do. You did not distribute $\sin x$ into $x^2+1$ first, because that would not make differentiation simpler here. You also did not try to combine the two terms. They are different kinds of terms, and addition is the correct final structure.

Write both terms first
If you simplify before you have $f'g+fg'$ fully written, you are more likely to drop a factor or accidentally differentiate the wrong piece twice.

Common patterns and simplifications

A lot of product rule practice is recognizing the same few pairings.

Patterns you will see constantly

• polynomial times trig, like $x\sin x$ or $(x^2+1)\cos x$
• polynomial times exponential, like $(3x-5)e^x$
• polynomial times log, like $x\ln x$

These derivatives often look messy at first, but they frequently simplify by factoring after differentiating. Factoring is not required, but it can make your answer easier to read and easier to use later.

Work through a few before-and-after examples and notice what the factor is in each case.

A good sign you can factor is when both terms share an obvious piece like $\sin x$, $e^x$, or a common polynomial factor.

Product rule vs chain rule vs quotient rule

Use the product rule when your function is a multiplication of two functions of $x$. Use the chain rule when one function is inside another. Use the quotient rule when it is a division, although rewriting as a product with a negative exponent is often cleaner.

If you want a quick, testable way to decide, this is it.

If the outermost structure is multiplication, it is product rule. If the outermost structure is composition, it is chain rule. If the outermost structure is a fraction, it is quotient rule or rewrite-first.

A decision table makes this feel automatic because it anchors the decision to what the expression looks like, not to what it reminds you of.

One fast self-check

After you differentiate a product, your result should be a sum of two terms. If you got a single term, you probably multiplied derivatives or accidentally combined terms that should not combine.

Each factor gets a turn

The mental model that sticks is each factor gets a turn. Differentiate one factor, keep the other alone, then swap.

It scales immediately. For three factors $fgh$, the derivative is still the same idea, just with three turns:

$$(fgh)'=f'gh+fg'h+fgh'.$$

No new trick is needed, just the same turn-taking pattern. When the factors themselves are complicated, you start mixing rules. For example, one factor might require the chain rule when you differentiate it, but the overall structure is still product rule turn-taking. The cleanest habit is to decide the outermost structure first, then apply inner rules only when you arrive at a factor you are differentiating.