Can it be concluded from $$f^{-1'}(x) = \frac{1}{f'(f^{-1}(x))}$$ that $$f'(x) = \frac{1}{f^{-1'}(f(x))}?$$ The reason I ask is because I have a function that I don't know the derivative of very easily, but I know its inverse and I know the derivative of its inverse, and this is about the only time I've ever used the inverse function theorem so I just want to make sure it works.
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Alternatively you could work the formula fairly quickly, simply differentiate both sides of :$$f(f^{-1}(x)) = x$$ – AgentS Mar 13 '18 at 06:15
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Well I'm looking specifically for more of a "yes" or "no" and what you just wrote doesn't help me with that. Even if I went through that work I would still only be trusting my own interpretation rather than the consensus of the people who use this system of reasoning. I would type 1+1 into my calculator just to make sure it's actually 2, and I've done that before. – John Joe Mar 13 '18 at 06:17
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In going from your first formula to second, you're simply replacing $f^{-1}$ by $f$, I don't see any issues with that. – AgentS Mar 13 '18 at 06:20
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Great, thank you. – John Joe Mar 13 '18 at 06:20
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Hopefully you typed 1+1 into calculator to make sure your calculator was working fine, not your arithmetic skills ;) – AgentS Mar 13 '18 at 06:25
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https://math.stackexchange.com/questions/315835/prove-the-relation-involving-derivative-of-inverse-of-a-function – user577215664 Mar 13 '18 at 06:25