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.
Differentiation
Taking the example of an implicit relation given above x2 + y2 = 81
Differentiating term by term (it is clearer if the dy⁄dx notation is used for implicit differentiation):
2x + 2y . dy⁄dx = 0
(y2 is differentiated using the Chain Rule − terms with y are differentiated and then multiplied by dy⁄dx)
Now make dy⁄dx 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.
Example
Differentiate 3x + 2xy = x2
3 + (2x . dy⁄dx + y . 2) = 2x
3 + 2x.dy⁄dx + 2y = 2x
re-arranging
2x. dy⁄dx = 2x − 2y − 3
