Cambridge International AS and A Level Mathematics Pure Mathematics 1

The chain rule

P1^

5

The chain rule

●?^ Whatinformationisgivenbyd

d

andd

d

Whatinforma

V

h

h

t

?

ttionisgivenbyd

d

d

d

V

h

h

t

×?

How would you differentiate an expression like

yx=+^2 1?

Your first thought may be to write it as y =^ (x^2 + 1)

(^12)
and then get rid of the
brackets, but that is not possible in this case because the power^12 is not a positive
integer. Instead you need to think of the expression as a composite function, a
‘function of a function’.
You have already met composite functions in Chapter 4, using the notation
g[f(x)] or gf(x).
In this chapter we call the first function to be applied u(x), or just u, rather than
f(x).
In this case, u = x^2 + 1
and y = u = u
(^12)
.
This is now in a form which you can differentiate using the chain rule.
Differentiating a composite function
To find
d
d
y
x for a function of a function, you consider the effect of a small change
in x on the two variables, y and u, as follows. A small change δx in x leads to a
small change δu in u and a corresponding small change δy in y, and by simple
algebra,
δ
δ
δ
δ
δ
δ
y
x
y
u
u
x

=×.

K

Volume 9