Saturday, 15 November 2014

calculus - Prove the relation involving derivative of inverse of a function



I want to prove the following result:



f1(x)=1f(f1(x))




Is simple application of chain rule a valid proof of it?



i.e. f(f1(x))=xdf(f1(x))dx=1 and hence the result. Or is this not a standard proof? Is there additional conditions necessary, expect of course that the function is bijective, or that the inverse exists.


Answer



The question is: Why is f1 differentiable? In the theorem you probably learned in a lecture this isn't assumed. However if you assume that f1 and f are differentiable at and all the expressions make sense f1 should be a function and f0 (both at least locally).



But nevertheless, with the chainrule the formula is easy to remember.


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