Differentiation of Implicit Relations

An explicit function is one where one variable is written in terms of the other such as y = 3x2 − 2x + 5

Expressions of this type can be differentiated term by term. e.g. y ' = 6x -2

An implicit relation is one where it is difficult to express one variable in terms of the other such as x2 + y2 = 81

This is the equation of a circle, centre (0, 0). It has a gradient at all points between -9 and +9, therefore it can be differentiated.


Taking the example of an implicit relation given above x2 + y2 = 81

Differentiating term by term (it is clearer if the dydx notation is used for implicit differentiation):

2x + 2y . dydx = 0              

 (yis differentiated using the Chain Rule − terms with y are differentiated and then multiplied by dydx)

Now make dydx the subject of the relation.


Only functions and relations involving xy and y2 need to be implicitly differentiated at this level.

Implicit Differentiation involving the Product rule

Sometimes there will be terms in implicit relations that need to be differentiated using the Product Rule.


Differentiate 3x + 2xy = x2

3 + (2x . dydx + y . 2) = 2x

3 + 2x.dydx + 2y = 2x


2x. dydx = 2x − 2y − 3