Quotient Rule

The quotient rule is a formula that allows you to differentiate a quotient of two functions
- i.e. one function divided by another
If $y = \frac{u}{v}$ where u and v are functions of x then the quotient rule is:

\frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}

In function notation, if $f (x) = \frac{g (x)}{h (x)}$ then the quotient rule can be written as:

f' (x) = \frac{h (x) g' (x) - g (x) h' (x)}{(h {(x))}^{2}}

As with the product rule, ‘dash notation’ may be used to make remembering it easier

y = \frac{u}{v}

y' = \frac{v u' - u v'}{v^{2}}

The quotient rule is used when trying to differentiate a fraction where both the numerator and denominator are functions of $x$
- if the numerator is a constant, negative powers can be used
- if the denominator is a constant, treat it as a factor of the expression\

Make it clear what $u, v, u'$ and $v'$ are
- arranging them in a square can help
  - opposite diagonals match up (like they do for product rule)

STEP 1

Identify the two functions,

u

and

v

Differentiate both

u

and

v

with respect to

x

to find

u'

and

v'

STEP 2

Obtain

\frac{d y}{d x}

by applying the quotient rule formula

\frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}

Be careful using the formula – because of the minus sign in the numerator, the order of the functions is important

Simplify the answer if straightforward or if the question requires a particular form

Quotient Rule Eg, AS & A Level Maths revision notes

The quotient rule formula is not on the list of formulas page – you have to memorise it
- however if you do forget it in an exam, you could rewrite $y = \frac{u}{v}$ as $y = u \times v^{- 1}$ and then use the product rule
- (the quotient rule is really doing exactly this)
Be careful using the formula – because of the minus sign in the numerator the order of the functions is important!

Quotient Rule Example, AS & A Level Maths revision notes

Quotient Rule (Cambridge O Level Additional Maths)