If we have
as a function of
it is quite easy to find
We often need to find
when
is a function ofor there are several occurrence of bothand
In these cases we need to differentiate implicitly. We shall start with a simple case.
Differentiate
We can differentiate both sides with respect to
obtaining 1 on the left hand side but when we differentiate the right hand side we must remember that we are differentiating a function of
with respect to
and so must use the chain rule, to get in this case
Hence 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 in
and factorise.
Example If
find
Differentiate each term with respect to
When we come to differentiating
we have to differentiate a product, so use the product rule obtaining
Hence we obtain:
