2.8 The Product Rule

Learning objectives

—FUN-3.B—Calculate derivatives of products and quotients of differentiable functions• FUN-3.B Calculate derivatives of products and quotients of differentiable functions.• FUN-3.B.1 Derivatives of products of differentiable functions can be found using the product rule.• Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

For a product $y=u(x)v(x)$ , the derivative has two contributions: first let $u$ change while $v$ stays as it is, then let $v$ change while $u$ stays as it is.

\frac{d}{dx}[u(x)v(x)]=u'(x)v(x)+u(x)v'(x)

Label the factors $u$ and $v$ .
Find $u'$ and $v'$ separately.
Form $u'v+uv'$ , keeping one original factor in each term.
Simplify or factor only after both terms are present.

For $y=x^2e^x$ , take $u=x^2$ and $v=e^x$ , so $u'=2x$ and $v'=e^x$ . Then $y'=(2x)e^x+x^2(e^x)=e^x(x^2+2x).$ Both terms are needed because both factors vary with $x$ .

In general, $(uv)'\ne u'v'$ . If one factor is a true constant, its derivative contribution is zero and the product rule reduces to the constant-multiple rule.

Next Topic

—2.9 The Quotient RuleRead