National 6 Mathematics
Pupil Revision Booklet
Differentiation
Please complete the questions and mark
using the answers. If you are stuck please ask
your teacher or drop into a supported study.
Differentiation - Part 1
1.
Differentiate each of the following with respect to x :
(a)
2.
3x 2 6x
(b)
x4 2x1 4
(2x 3)2
(c)
Differentiate each of the following functions with respect to the relevant variable :
(a)
(b) g(x) 3x2 
f (x) x 3 (x x2 )
1
x
(d)
f (t) t 2 (t 2  t 2 )
1
3
(e)
1
g( p)

p
(g)
(h)
h (x) 
(f)
h (u) 
3
( p 3  p 2 3 )
1
1
 1
x
3x2
1
u 
2 u

5
2
g(x)  x 2x
x4
(c)
f (v)  5v 2
v

1 t 2 t 2 
h (t )  

t  t 
3
(i)



3.
Calculate the rate of change of :
(a)
4.
f (x) x3 2x 2
at
x 2
(b)
h (t) (t 1)(2t 3)
at
t 1
The amount of pressure P (pounds per square inch) within a cylinder varies with
time t (milliseconds) according to the formula P (t)  400t  50t2 .
(a)
Calculate the rate of change of P when t 2 .
(b)
Calculate how fast P is changing when t 4 .
(c)
How is the rate of P changing when t 5 ? Comment.
5.
(a)
Find the equation of the tangent to the curve with equation
the point where x 1 .
(b)
Find the equation of the tangent to the curve
y 3x2  2x at


y 
1
x at the point where
x
x  12 .
6.
A function f is given by f (x)  3x2 2x3 . Determine the interval on which f is
increasing.
7.
Find the stationary points of the curve y  x2 (12 4x 3x2 ) and determine their
nature.
8.
9.
A curve has as its equation y  x ( 2  x )2 .
(a)
Find the stationary points of the curve and determine the nature of each.
(b)
Write down the coordinates of the points where the curve meets the
coordinate axes.
(c)
Sketch the curve.
A rectangular field is x metres long. It is enclosed by 200 metres
of fencing.
(a)
Find an expression in terms of x for the breadth of the field .
(b)
Hence find the greatest area which can be enclosed with the 200 metres of
fencing.
10.
An open cistern with a square base and vertical sides is to have a
capacity of 4000 cubic feet.
h
(a)
Taking the length of the square base to be x feet , find an
expression for the height h in terms of x .
x
x
(b)
Hence show that the surface area, A square feet, of the cistern
can be written in the form A(x)  x2
(c)
+ 16000

x
Find the dimensions of the cistern so that the cost of cladding it in lead sheet
will be a minimum.
Differentiation - Part 2
1.
Differentiate
x 2 1
,
x
with respect to x ,
expressing your answer with positive indices.
2.
Find
f (x) when
f (x) 
x 3 6 x
x
3.
.
2
The diagram below shows the parabola with equation y 8x  3x2 and the
line which is a tangent to the curve at the point T(1,5).
y
T(1,5)

o
x
y  8x 3x
2
Find the size of the angle marked  , to the nearest degree.
4.
The sketch shows part of the graph of y x3 4 x .
y
y x 3  4 x

The tangent at the point where x 1 is also shown.
o
(a)
Find the equation of the tangent.
(b)
Given that the axes are not to scale, calculate the true
anti-clockwise angle between this tangent and the
x - axis, giving your answer correct to the nearest degree.
1
x
5.
Show that the function f (x)  4(1 2x)3 is decreasing for all values
of x , except x  12 .
6.
Show that the curves with equations
y  x2 8x 3
and
y 1 4x x2
touch each other at a single point, and find the equation of the common tangent at
this point.
7.
A function is given as
f (x)  5x5  10x3  9x and is defined on the
set of real numbers.
(a)
Show that the derivative of this function can be expressed in
the form f (x)  ( ax 2  b )2 and write down the values of a and b .
(b)
Hence explain why this function has no stationary points and is in fact
increasing for all values of x .
8.
A tank has to be designed for the top of a water tower. It must be rectangular in
shape and open at the top.
Its base sides are to be in the ratio 1 : 3.
(a)
Taking the shorter of the base sides to be x ,
show that the tanks surface area (A) can be
expressed as
h
x
A  3x2  8xh ,
where h is the height of the tank.
(b)
Given that the tank has to constructed from 144 sq. metres
of steel plate construct a formula for h in terms of x .
(c)
Hence, or otherwise, show that the volume of the tank, in terms of x , is given
as
V (x)  54x  98 x 3 .
(d)
Find the dimensions of the tank which will maximise its volume.
Differentiation - Part 1 Answers
1.
(a)
6x 6
(b)
4x3 2x2
2.
(a)
4x3 5x4
(b)
6x 3x4
(d )
5
t 2 2t
(e)
4 p 3 1 p 3
3
(c)
7
2
4
3
8x 12
(c)
(f )
3
v 2 v 2
100
(pressure is decreasing)
(h)
3.
(a)
4
(b)
3
4.
(a)
200
(b)
0
5.
(a)
6.
f is increasing when 0 < x < 1 .
7.
(-2,32) , Max. , (0,0) , Min. , (1,5) , Max.
8.
(a)
( 23 ,
(b)
(0,0)
y 8x 3
32
27
) , Max.
(b)
,
and ( 2 , 0 )
(c)
9.
10.
(a)
100 x
(b)
2500 square metres
(a)
h  4000
x2
3
(c)
3
u 2 14 u 2
1 1 2
2
1 4x3
1
1
1
2
(i)
(g)
5
2
x2 23 x3
6x 2y 8
( 2 , 0) , Min.
t
(b)
Proof
(c)
20 by 20 by 10 (since x 20 and h 10)
Differentiation - Part 2 Answers
1.
3
1
x 12 
2
2x
3
2
2.
f (x) 1 9x
3.
63
(a)
5.
proof
6.
T(-1,-4)
7.
(a)
(a)
2
y  8x 5
4.
8.
5

83
(b)
, y = 6x + 2
a = 5 and b = 3
proof (b)
(b)
144 3x 2
h 
8x
Explanation f (x) must be > 0 since a perfect
square  function is never decreasing
(c) proof
(d)
4cm by 12cm by 3cm