Math101 · Independent Ontario learning library
Math101Product Rule
The product rule differentiates two changing factors by adding the change from each factor while the other is held in place.
The derivative of a product is not the product of the derivatives; each factor contributes to the total change.
The rule
If $u(x)$ and $v(x)$ are differentiable, then
A common memory phrase is “first derivative times second, plus first times second derivative.” Clear function labels are safer than relying on wording alone.
Worked example
Expanding the original first and differentiating term by term gives the same result.
Product rule with the chain rule
Different factors may require other rules internally. For
the product rule handles the two factors and the chain rule differentiates $(3x+1)^5$:
Rule selection follows layers of structure.
Evaluating without fully simplifying
To find a slope at $x=a$, it may be fastest to substitute into
without expanding the derivative. A table can supply these four values even when formulas are unavailable.
For instance, if $u(2)=3$, $u'(2)=5$, $v(2)=-1$, and $v'(2)=4$, then $(uv)'(2)=5(-1)+3(4)=7$.
Common mistakes
Writing $(uv)'=u'v'$. The two contribution terms are required.
Differentiating only one factor. Both change with $x$.
Forgetting parentheses around a full factor. Preserve its structure.
Missing an inner chain-rule factor. Product rule handles the outside product only.
Expanding when it creates avoidable errors. Simplify strategically, not automatically.
Quick self-check
- Are there two nonconstant factors?
- Have I labelled $u,v,u',v'$ correctly?
- Does the result contain $u'v+uv'$?
- Do any factors need chain or other rules internally?
- Can the derivative be factored for later analysis?
- Do units or a second method confirm the result?
