Using the Quotient Rule | CAIE 9709 Pure Mathematics 2 Notes

Pure Mathematics 2

Mathematics9709

Protect the numerator order

Key idea

y=u/v

, then

\frac{dy}{dx}=\frac{vu'-uv'}{v^2}

Choose the complete numerator as $u$ .
Choose the complete denominator as $v$ .
Find $vu'$ and $uv'$ on separate lines.
Form $(vu'-uv')/v^2$ , then simplify.

Show both numerator pieces

Worked example

A quotient with a composite numerator

y=\frac{(2x-1)^2}{x+3}

u' = 4(2x-1),\qquad v'=1

vu'=4(x+3)(2x-1)

uv'=(2x-1)^2

y'=\frac{vu'-uv'}{v^2}

y'=\frac{(2x-1)(2x+13)}{(x+3)^2}

Common mistake

Squaring the denominator is part of the rule. It does not mean that the original denominator was squared before differentiation.

Choose a useful rewrite

$f(x)/x^n$ can sometimes be rewritten as $f(x)x^{-n}$ and differentiated with the product rule. Choose the route that keeps the working shorter and clearer.

Your turnIndependent check [5 marks]

For $y=(x^2+1)/(x-2)$ , find $dy/dx$ and then the gradient at $x=3$ .

Show worked answer

vu'=2x(x-2)

uv'=x^2+1

y'=\frac{vu'-uv'}{v^2}

y'=\frac{x^2-4x-1}{(x-2)^2}

Answer

\left.\frac{dy}{dx}\right|_{x=3}=-4