Cambridge International AS and A Level Mathematics Pure Mathematics 1

P1^

5

The

(^) second
(^) derivative
ACTIVITY 5.7 Sketch the graph of the gradient of d
d
y
x
against x for the function illustrated in
figure 5.25. Do this by tracing the two graphs shown in figure 5.25, and extending
the dashed lines downwards on to a third set of axes.
You can see that P is a maximum point and Q is a minimum point. What can
you say about the gradient of d
d
y
x
at these points: is it positive, negative or zero?
The gradient of any point on the curve of d
d
y
x
is given by
d
d
d
x d
y
x







. This is written

as d

d

2

2

y

x

or f′′(x), and is called the second derivative. It is found by differentiating

the function a second time.

!^ The second derivative,

d

d

2

2

y

x

, is not the same as

d

d

y

x









2

.

EXAMPLE 5.14 Given that y = x^5 + 2 x, find d
d

2

2

y

x

.

SOLUTION

d

d

d

d

y

x

x

y

x

x

=+

=

52

20

4

2

2

(^3).
Using the second derivative
You can use the second derivative to identify the nature of a stationary point,
instead of looking at the sign of d
d
y
x
just either side of it.
Stationary points
Notice that at P, d
d
y
x
= 0 and d
d
2
2
y
x
< 0. This tells you that the gradient, d
d
y
x
, is zero
and decreasing. It must be going from positive to negative, so P is a maximum
point (see figure 5.26).
At Q, d
d
y
x
= 0 and d
d
2
2
y
x

0. This tells you that the gradient, d
   d
   y
   x
   , is zero and
   increasing. It must be going from negative to positive, so Q is a minimum point
   (see figure 5.27).