DN1.6: PRODUCT RULE
The ‘product rule’ is used when we want to differentiate the product of two functions
If
f(x) = u(x).v(x)
then f'(x) = u(x).v'(x) + u'(x).v(x)
which is often abbreviated to
=
y′ uv′ + u ′v
Examples
1.
Find the derivative of f(x) = (x + 3)6 (2x – 1)
Let u = (x+3)6
and
let v = 2x - 1
u' = 6(x+3)5
v' = 2
Then y' = uv' + u'v
= (x + 3)6.2 + 6(x + 3)5(2x – 1)
= 2(x + 3)6 + 6(x + 3)5(2x – 1)
...and this could (but does not have to be) simplified further…..
= 2(x + 3)5 [(x + 3) + 3(2x – 1)]
[by factorizing]
= 2(x + 3)5 (7x)
= 14x(x + 3)5
2. Differentiate e sin2x
x
Let u = e
x
and v = sin2x
u ′ = e x and v′ = 2cos2x
Then
[using the chain rule]
y ′ = u v′ + u ′ v
x
x
= e .2cos2x + e sin2x
x
x
= 2e cos2x + e sin2x
DN 1.6: Product Rule
Page 1 of 2
June 2012
Exercises
1. Use the product rule to differentiate the following
a) y = (x – 2)(6x + 7)
b) f(x) = (2x2 + 4)(x5 + 4x2 – 2)
[simplify as far as possible]
c) y = (
x - 1)(x2 + 1)
d) y = (x3 – 4x +
x )(3x4 + 2)
[simplify as far as possible]
e) f(x) =
x − 1 (x + 1)2
f) f(x) = (x2 – 1)(x2 + 1)
[simplify as far as possible]
2.
Find the derivative of
2
2
a) y = e x tanx
b) y = x log e (x )
c) y = sinxcosx
d) y = xsinx
e) y = (2 – x)tan3x
ex
f) y =
x
[Hint:
1
x
= x -1]
Answers
1. (a) 12x – 5
(b) (2x2 + 4)(5x4 + 8x) + 4x(x5+ 4x2 – 2)
(c)
3
5
1
x 2 - 2x +
1
2
2x 2
1
(d) (x3 – 4x + x 2 )(12x3) + (3x2 – 4 +
1
1
)(3x4 + 2)
2x 2
x −1 +
(e) 2(x + 1)
( x + 1)
2
2 x −1
3
(f) 4x
2. (a)
(b)
x
x
e tanx + e sec2x
2
2x + 2x log e (x )
2
2
(c) cos x - sin x
(d) xcosx + sinx
(e) 3(2-x)sec2 3x – tan3x
(f)
e
x
x
−
e
x
x2
DN 1.6: Product Rule
Page 2 of 2
June 2012