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Topic: Derivative help (Read 7375 times) 

Wardub
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Derivative help
« on: Jul 21^{st}, 2010, 10:04am » 
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(A^1)' (x) = 1/(A'(A^1(x)) That should read the derivative of A inverse. I'm trying to follow a proof and can't understand how he got this. It seems like it should involve the chain rule. Can someone help me break it down step by step? Thanks.


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0.999...
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Re: Derivative help
« Reply #1 on: Jul 21^{st}, 2010, 10:18am » 
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Assuming the existence of an inverse of A, we have the equation, A(A^{1}(x)) = x . Now, we implicitly differentiate w.r.t. x and indeed the chain rule implies that (A^{1})'(x)*A'(A^{1}(x)) = 1 . Hence the result. Visually, since the graph of the inverse function A^{1} is a reflection of the initial function A across y = x, if at (a,b) A has slope dy/dx then at (b,a) A^{1} has slope dx/dy.


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