Product Rule

The product rule is a formula that allows you to differentiate a product of two functions
If $y = u \times v$ where u and v are functions of x then the product rule is:

\frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}

In function notation, if $f (x) = g (x) \times h (x)$ then the product rule can be written as:

f' (x) = g (x) h' (x) + h (x) g' (x)

The easiest way to remember the product rule is, for $y = u \times v$ where u and v are functions of x:

y' = u v' + v u'

The product rule is used when we are trying to differentiate the product of two functions
- These can easily be confused with composite functions (see chain rule)
  - $\sin (\cos x)$ is a composite function, “sin of cos of $x$ ”
  - $\sin x \cos x$ is a product, “sin x times cos $x$ ”

Make it clear what $u, v, u'$ and $v'$ are
- arranging them in a square can help
  - opposite diagonals match up

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 product rule formula

\frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}

Simplify the answer if straightforward to do so or if the question requires a particular form

The product rule formula is not on the list of formulas page – make sure you know it
Don't confuse the product of two functions with a composite function:
- The product of two functions is two functions multiplied together
- A composite function is a function of a function

Product Rule Prod Comp Illustr, AS & A Level Maths revision notes

Product Rule Example, A Level & AS Level Pure Maths Revision Notes

Product Rule (CIE IGCSE Additional Maths)