Further Differentiation (Cambridge (CIE) A Level Maths: Pure 3): Flashcards

Exam code: 9709

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FrontQuotient Rule
For $y = \frac{u}{v}$ , where $u$ and $v$ are functions of $x$ , complete the quotient rule:
$\frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{______}$

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For $y = \frac{u}{v}$ , where $u$ and $v$ are functions of $x$ , complete the quotient rule:
$\frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{______}$
The completed rule is:
$\frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$
The denominator is the bottom function squared, not the derivative of anything, and the whole formula is given in the formulae booklet.
Why does the order of the two terms matter in the quotient rule?
Because of the minus sign in the numerator: swapping the terms reverses the sign of the whole answer.
The term beginning with $v$ , the bottom function, is the one that comes first.
How can you recognise a quotient rule question written as $g (x) {(h (x))}^{- 1}$ ?
A negative power applied to a whole function is a division in disguise, since $g (x) {(h (x))}^{- 1} = \frac{g (x)}{h (x)}$ .
It can be done with the product and chain rules instead, but the quotient rule is usually quicker.
Differentiate $y = \frac{\sin x}{x}$ .
Taking $u = \sin x$ and $v = x$ :
$\frac{d y}{d x} = \frac{x \cos x - \sin x}{x^{2}}$
Answers from the quotient rule rarely simplify much, so leaving the result as a single fraction is normal.
True or False?
Every quotient has to be differentiated using the quotient rule.
False.
A quotient that simplifies should be simplified first: $\frac{x^{3} + x}{x}$ is just $x^{2} + 1$ , which differentiates in one line.
The rule is for quotients that cannot be reduced to a sum of simpler terms.

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