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Implicit Differentiation

If we haveas a function ofit is quite easy to findWe often need to find whenis a function ofor there are several occurrence of bothandIn these cases we need to differentiate implicitly. We shall start with a simple case.

Differentiate

We can differentiate both sides with respect toobtaining 1 on the left hand side but when we differentiate the right hand side we must remember that we are differentiating a function ofwith respect toand so must use the chain rule, to get in this caseHence differentiating both sides with respect togives us

If we are to express {dy} over {dx} in terms ofwe can do it in this case (it is not always possible) by using

More complicated expressions may have several occurrences oforand may require us to group terms inand factorise.

Example Iffind

Differentiate each term with respect to

When we come to differentiatingwe have to differentiate a product, so use the product rule obtainingHence we obtain: