2.9 The Quotient Rule

Learning objectives

—FUN-3.B—Calculate derivatives of products and quotients of differentiable functions—Topic 2.9• FUN-3.B Calculate derivatives of products and quotients of differentiable functions.• FUN-3.B.2 Derivatives of quotients of differentiable functions can be found using the quotient rule.• Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

For $y=\dfrac{u(x)}{v(x)}$ , both $u$ and $v$ may change. The quotient rule combines those changes in a fixed order and applies wherever $u$ and $v$ are differentiable and $v(x)\ne0$ .

\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2}

Label the numerator $u$ and denominator $v$ .
Find $u'$ and $v'$ separately.
Build the numerator as $u'v-uv'$ in that order.
Divide by $v^2$ , then simplify without changing the original domain.

For $y=\dfrac{\sin x}{x}$ , let $u=\sin x$ and $v=x$ , so $u'=\cos x$ and $v'=1$ . Then $y'=\frac{x\cos x-(\sin x)(1)}{x^2}=\frac{x\cos x-\sin x}{x^2},\qquad x\ne0.$

In general, $\left(\dfrac{u}{v}\right)'\ne\dfrac{u'}{v'}$ . Do not reverse the subtraction, forget the square on $v$ , or include inputs where the original denominator equals zero.

Next Topic

—2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant FunctionsRead