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.
A Level Maths Notes: C3 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 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:
