An **explicit **function is one where one variable is written in terms of the other such as y = 3x^{2} − 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 x^{2} + y^{2} = 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 x^{2} + y^{2} = 81

Differentiating term by term (it is clearer if the ^{dy}⁄_{dx} notation is used for implicit differentiation):

2x + 2y . ^{dy}⁄_{dx} = 0

(y^{2 }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 y^{2} 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 = x^{2}

3 + (2x .

^{dy}⁄_{dx}+ y . 2) = 2x3 + 2x.

^{dy}⁄_{dx}+ 2y = 2xre-arranging

2x.

^{dy}⁄_{dx}= 2x − 2y − 3