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: