The Product Rule
Introduced| English |
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| Product Rule/ˈprɒdʌkt ruːl/ |
Why products need their own rule
- Tempting but wrong: the derivative of a product is not the product of the derivatives.
- $\dfrac{d}{dx}[x\cdot x]=\dfrac{d}{dx}[x^2]=2x$, but $f'g'=1\cdot1=1$. They disagree.
- Products get their own tool: the Product Rule 乘积法则.
- It accounts for both factors changing at once.
The derivative of $f\cdot g$ equals $f'\cdot g'$.
It is $f'g+fg'$, not $f'g'$.
The rule
- For a product of two functions:
-
$$\frac{d}{dx}\big[f\,g\big]=f'\,g+f\,g'$$
- In words: "derivative of the first times the second, plus the first times derivative of the second."
- Each term differentiates one factor and leaves the other alone; then you add.
The Product Rule states $\dfrac{d}{dx}[fg]=$
Derivative of first times second, plus first times derivative of second.
A clean worked pattern
- Differentiate $y=x^2\,e^x$. Let $f=x^2$ and $g=e^x$.
- $f'=2x$, $g'=e^x$.
- $y'=f'g+fg'=2x\,e^x+x^2\,e^x$.
- Factor if asked: $y'=x e^x(2+x)$.
A product's changing slope
y = ax² + bx
A product like $x^2 e^x$ has a slope that mixes both factors — the Product Rule captures both changing at once.
Differentiate $y=x^2 e^x$.
$f'g+fg'=2x e^x+x^2 e^x$.
For $y=x\sin x$, use the product rule to find $y'$ at $x=0$. (Recall $\sin 0=0$, $\cos 0=1$.)
$y'=\sin x+x\cos x$; at $0$: $0+0\cdot1=0$.
When you actually need it
- If a product can be multiplied out first, do that — it may be easier ($x^2\cdot x^3=x^5\Rightarrow5x^4$).
- But when the factors won't simplify (like $x^2\sin x$ or $e^x\ln x$), the rule is required.
- Combine it freely with the power and elementary derivatives for each factor.
- Label $f,f',g,g'$ first — organized bookkeeping prevents mistakes.
For $y=x^2\cdot x^3$, the smartest first move is to...
Simplifying to $x^5$ first is easier: $5x^4$. Use the Product Rule only when factors won't combine.
In each term of the Product Rule, which is true?
One factor per term is differentiated; the two terms are summed. Differentiating both in one term is the classic error.
The Product Rule is $f'g+fg'$, not $f'g'$. Both terms are needed, and each keeps the other factor undifferentiated. A very common slip is to differentiate both factors in the same term — don't. Exactly one factor is differentiated per term.
Differentiate $y=(3x^2)(\sin x)$.
- $f=3x^2\Rightarrow f'=6x$; $\quad g=\sin x\Rightarrow g'=\cos x$.
- $y'=f'g+fg'=6x\sin x+3x^2\cos x$.
- Neither factor could be simplified away, so the Product Rule was necessary.
The Product Rule: $\frac{d}{dx}[fg]=f'g+fg'$ — differentiate one factor at a time and add. Use it whenever a product of differentiable functions can't be simplified first. It is not $f'g'$; each term differentiates exactly one factor.