Product Rule for Differentiation

Let y = u. v, where both u and v are differentiable functions of x.Let y = u. v, where both u and v are differentiable functions of x.

or (u.v)' = u.v' + v .u'

It can be remembered as:

first function x derivative of second function + second function x derivative of first function'

Remark:

If we divide by uv in the formula ((u.v)' = u.v' + v .u'), we have

It can be generalised for derivative of the product of more than two functions as given below:

Example 1:

Differentiate xe^x from the first principles (that is using definition of derivative).

Suggested answer:

Let y = f(x) = xe^x

= xe^x (1) + e^x(1)

Example 2:

Differentiate the following function using product rule.

y = (3 sec x - 4 cosec x) (2 sin x + 5 cos x)

Suggested answer:

= (3 sec x - 4 cosec x) (2 cos x - 5 sin x)

+ (2 sin x + 5 cos x) (3 sec x.tan x + 4 cosecx cot x)

= 6 sec x. cos x - 15 sec x. sin x - 8 cosec x. cos x

+ 20 cosec x. sin x + 6 sin x sec x. tan x + 8 sin x. cosec x . cot x + 15 cos x. sec x. tan x + 20 cos x. cosec x. cot x

= 6 - 15 tan x - 8 cot x + 20 + 6 tan²x + 8 cot x + 15 tan x

+ 20 cot²x

= (6 + 6 tan²x) + (20 + 20 cot²x)

= 6(1+tan²x) + 20 (1+cot²x)

= 6 sec²x + 20 cosec²x.

Example 3:

Differentiate y = (x+2) (x+3) by using the formula

Suggested answer:

Let u = x + 2, v = x + 3

= 2x + 5