The product rule and quotient rule are used when a function is built from multiplication or division
of two changing expressions.
1. Product rule
If:
\[
f(x)=u(x)v(x)
\]
then:
\[
\boxed{f'(x)=u'(x)v(x)+u(x)v'(x)}
\]
The important idea is that only one factor is differentiated at a time.
2. Product rule example
Differentiate:
\[
f(x)=x^2\sin x
\]
Let:
\[
u=x^2,\qquad v=\sin x
\]
Then:
\[
u'=2x,\qquad v'=\cos x
\]
Apply the product rule:
\[
f'(x)=u'v+uv'
\]
\[
f'(x)=2x\sin x+x^2\cos x
\]
Final answer:
\[
\boxed{f'(x)=2x\sin x+x^2\cos x}
\]
3. Quotient rule
If:
\[
f(x)=\frac{u(x)}{v(x)}
\]
then:
\[
\boxed{
f'(x)=\frac{u'(x)v(x)-u(x)v'(x)}{[v(x)]^2}
}
\]
The subtraction order matters: \(u'v-uv'\), not \(uv'-u'v\).
4. Quotient rule example
Differentiate:
\[
f(x)=\frac{x^2+1}{\sin x}
\]
Let:
\[
u=x^2+1,\qquad v=\sin x
\]
Then:
\[
u'=2x,\qquad v'=\cos x
\]
Apply the quotient rule:
\[
f'(x)=\frac{u'v-uv'}{v^2}
\]
\[
f'(x)=\frac{2x\sin x-(x^2+1)\cos x}{\sin^2 x}
\]
5. Why the product rule is not just \(u'v'\)
A common mistake is:
\[
(uv)'=u'v'
\]
This is false. Both factors can change, so the derivative has two terms:
\[
(uv)'=u'v+uv'
\]
6. Why the quotient rule has \(v^2\)
The quotient rule can be derived by writing:
\[
\frac{u}{v}=u\cdot v^{-1}
\]
Use the product rule:
\[
\frac{d}{dx}(uv^{-1})=u'v^{-1}+u(-1)v^{-2}v'
\]
Rewrite with a common denominator:
\[
\frac{u'}{v}-\frac{uv'}{v^2}
=
\frac{u'v-uv'}{v^2}
\]
7. Choosing the correct rule
- Use the product rule when the main operation is multiplication.
- Use the quotient rule when the main operation is division.
- Use the chain rule inside \(u\) or \(v\) when a factor is composite.
- Use the power rule inside \(u\) or \(v\) when a factor is a power.
8. Rule-specific checklist
| Rule |
Setup |
Formula |
Common mistake |
| Product rule |
\(f=uv\) |
\(f'=u'v+uv'\) |
Writing \(u'v'\) only. |
| Quotient rule |
\(f=\dfrac{u}{v}\) |
\(f'=\dfrac{u'v-uv'}{v^2}\) |
Using plus instead of minus. |
| Numerator |
\(u\) |
Differentiate to get \(u'\) |
Forgetting to multiply \(u'\) by \(v\). |
| Denominator |
\(v\) |
Differentiate to get \(v'\) |
Forgetting that the final denominator is \(v^2\). |
9. Common mistakes
- Using \(u'v'\) instead of \(u'v+uv'\).
- Leaving out the second term in the product rule.
- Writing \(u'v+uv'\) for a quotient.
- Using \(v\) instead of \(v^2\) in the quotient-rule denominator.
- Reversing the subtraction order in the quotient rule.
- Forgetting the chain rule inside \(u\) or \(v\).
10. Good practice strategy
For each problem, write four small pieces before combining:
\[
u,\quad v,\quad u',\quad v'
\]
Then substitute these pieces into the correct rule. This reduces sign errors and missing-term errors.
