Chapter 30

Differentiation of implicit

functions

30.1 Implicit functions

When an equation can be written in the formy=f(x)
it is said to be anexplicit functionofx.Examplesof
explicit functions include

y= 2 x^3 − 3 x+ 4 , y= 2 xlnx

andy=

3ex

cosx

In these examplesymay be differentiated with respect
toxby using standard derivatives, the product rule and
the quotient rule of differentiation respectively.
Sometimes with equations involving, say,yandx,
it is impossible to makeythe subject of the formula.
The equation is then called animplicit functionand
examples of such functions include
y^3 + 2 x^2 =y^2 −xand siny=x^2 + 2 xy.

30.2 Differentiating implicit functions

It is possible todifferentiate an implicit functionby
using thefunction of a function rule,whichmaybe
stated as

du

dx

=

du

dy

×

dy

dx

Thus, to differentiatey^3 with respect tox, the sub-

stitutionu=y^3 is made, from which,

du

dy

= 3 y^2. Hence,

d

dx

(y^3 )=( 3 y^2 )×

dy

dx

, by the functionofa functionrule.

A simple rule for differentiating an implicit function

is summarised as:

d

dx

[f(y)]=

d

dy

[f(y)]×

dy

dx

(1)

Problem 1. Differentiate the following functions

with respect tox:

(a) 2y^4 (b) sin3t.

(a) Letu= 2 y^4 , then, by the function of a function

rule:

du

dx

=

du

dy

×

dy

dx

=

d

dy

( 2 y^4 )×

dy

dx

= 8 y^3

dy

dx

(b) Letu=sin3t, then, by the function of a function

rule:

du

dx

=

du

dt

×

dt

dx

=

d

dt

(sin 3t)×

dt

dx

=3cos3t

dt

dx

Problem 2. Differentiate the following functions

with respect tox:

(a) 4 ln5y (b)

1

5

e^3 θ−^2

(a) Letu=4ln5y, then, by the function of a function

rule: