at a point where f and g are both zero. 0 over 0 is undefined so it
might be thought the derivative of f over g does not exist. Of ten
however the derivative does exist and we can evaluate it using
L'Hopital's Rule:
University Maths Notes: Calculus – L'Hopital's Rule
L'Hopital's Rule is used to differentiate a quotient of
two functions
at a point where f and g are both zero. 0 over 0 is undefined so it
might be thought the derivative of f over g does not exist. Of ten
however the derivative does exist and we can evaluate it using
L'Hopital's Rule:
If
and
then
if
and
are
not both zero.
L'Hopital's rule can be used repeatedly so if
are
all zero then
Proof: A simple proof can be constructed using Taylor
series for
and
around
the point
so
Now let
and
all the
and
higher order terms vanish. Hence
Example: Differentiate
at
so
L'Hopital' Rule applies.
