Let be an open interval and be a one-one differentiable function. Let be the range of and be the inverse function. Then, is differentiable at a point for , such that and in that case
Proof:
Since is one-one and continuous on is either strictly increasing or strictly decreasing. Let . Then, and since is one-one, for Let and . Note that since both and are continuous, if and only if . Thus,
be an open interval and 
be a one-one differentiable function. Let 
be the range of 
and 
be the inverse function. Then, 
is differentiable at a point 
for 
, such that 
and in that case 
is one-one and continuous on 
is either strictly increasing or strictly decreasing. Let 
. Then, 
and since 
is one-one, 
for 
Let 
and 
. Note that since both 
and 
are continuous, 
if and only if 
. Thus, 
Back
