Learning Enhancement Team
Worksheet:
The Product Rule
This worksheet has questions about using The Product Rule: the method of differentiating
product functions. Using the product rule is common in calculus problems. Before
attempting the questions below you should be able to differentiate basic functions and
understand what a product function is.
Differentiating
Basic Functions
study guide
More Complicated
Functions study guide
Product Rule
study guide
Model Answers
to this Sheet
If
y  uv
then
dy
dv
du
u
v
dx
dx
dx
1.
Find the derivatives of the following product functions using the suggested u and v.
(a)
y  x 5 sin x
(use u  x 5 and v  sin x )
(b)
y  3e x cos x
(use u  3e x and v  cos x )
(c)
y  sin3x cos5x 
(use u  sin3x  and v  cos5x  )
(d)
y
 t
cos t
5
(use u 
(e)
y
5
3x  5e x
6
(use u  3x  5 and v 
(f)
y
(g)
y  x 3  2.5x 2  0.3 ln x
sin 

 t
and v  cos t )
5
5 x
e )
6
(use u   1 and v  sin )
(use u  x 3  3.5x 2  0.3 and v  ln x )
2. Evaluate whether the following functions can be differentiated with respect to x using the
product rule or not. If you decide you can use the product rule, choose an appropriate u
and v and find the derivative.
(a)
y  3x 3 ln( 3x )
(b)
y  cosln x 
(c)
y  2 sin x cos x
(d)
y  3x  e x  x 2e x
(e)
y  x 3 ln t
(f)
y
3.
Find the gradient of the following curves at the given value of x:
(a)
y  x 3 x 2  x  2
(c)
x
y    cos x
 
for x  1
2
for x 

2
3x
5
 6 x 4  3 x 2 e 4 x
3x 2
for x  2
(b)
y  x ln x
(d)
y  x 2  1 e x for x  0
2
This worksheet is one of a series on
mathematics produced by the Dean of
Students' Office at the University of East
Anglia.
Scan the QR-code with a smartphone to go
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