The Inverse Function Rule
If
is continuous and one to one on an interval
then
exists and is continuous. Ifis differentiable then the following theorem is applicable:
Theorem
Suppose
is continuous and differentiable with
for all
Thenis one to one,is continuous and differentiable on
and
for all
Proof: Sincefor all
thenis one to one.
Suppose that
–is continuous sois a closed interval. Choose
and 
any sequence in
converging to
Let
for
is continuous sois continuous and
converges to
and sinceis one to one
for all
Sinceis differentiable
converges to
and sinceis one to one
for
Hence
converges to
Thusis differentiable and
Example
and
so
Example
and
so
