I want to prove the following result:
f−1′(x)=1f′(f−1(x))
Is simple application of chain rule a valid proof of it?
i.e. f(f−1(x))=x⟹df(f−1(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 f−1 differentiable? In the theorem you probably learned in a lecture this isn't assumed. However if you assume that f−1 and f are differentiable at and all the expressions make sense f−1 should be a function and f′≠0 (both at least locally).
But nevertheless, with the chainrule the formula is easy to remember.
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