that is continuousfor all The inverseof is whereits domainis all
and its rangeis We concludethat if is a functionwith domain and range and it is continuous
and one-to-oneon then its inverse is continuousand one-to-oneon the range of
Supposethat has a domain and a range If is differentiableand one-to-oneon then its inverse
is differentiableat any value in for which and
The formulaabovecan be writtenin a form that is easierto remember:
In addition,if on its domainis either or then has an inversefunction and
is differentiableat all valuesof in the rangeof In this case, is givenby the formulaabove.
The examplebelowillustratethis importanttheorem.
Example4:
In Example3, we weregiventhe polynomialfunction and we showedthat it is in-
vertable.Showthat it is differentiableand find the derivativeof its inverse.
Solution:
Since for all is differentiableat all valuesof To find the
derivativeof if we let then
So
and
Sincewe are unableto solvefor in termsof we leavethe answerabovein termsof Anotherway
of solvingthe problemis to use ImplicitDifferentiation: