C3 2016 2 Edexcel

2

(a)

By the quotient rule we have:

\begin{align*}\frac {\mathrm dy} {\mathrm dx} & = \frac {4(x)' (x^2 + 5) - 4x(2x)} {(x^2 + 5)^2} \\ & = \frac {4(x^2 + 5) - 8x^2} {(x^2 + 5)^2} \\ & = \frac {20 - 4x^2} {(x^2 + 5)^2}\end{align*}

Product rule then chain rule also works.

(b) We have:

\displaystyle \frac {\mathrm dy} {\mathrm dx} < 0

iff:

20 - 4x^2 < 0

(since (x^2 + 5)^2 > 0 for all x)

Therefore:

x^2 > 5

So:

\displaystyle x > \sqrt 5 or \displaystyle x < -\sqrt 5

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