Question

1. The chain rule states for (fog)(x) = h(x), h'(x) = f'(g(x))g'(x). (i) Using the chain rule and that y = g(x) = f-1(x), prove the Inverse Function Theorem (F-1)'(x) = Fitu). Explain or justify each step in your proof. (ii) Write a few sentences about how f'(x) corresponds to (f-1)'(x) graphically. (iii) Let f(x) be a non-linear function. If possible, find a function f such that f(4) = 2, (4-1)'(2) = If this task is impossible, explain why.

Answer 1

Suppose that firR" is continuaily differentiable Post DL in an open Then there is an open and fvw containing a, and det t'(a) to set & containing an open set w containing f(a) such that has continuous inveore file v which is differentinkte and for all 4 ew satisfies. (F' - [PPLE') is yog(x). 7'() bet d be lineae transformation Delay! Then & non-singular hence since det f'(a) to. Now (d'of) (a) = D(+1) f (as) odtla) - Aloof (a) ru identity lineae transformations if the theorom is true for d'ot, it is clearly true for of assume at outset' that d is identity. Thus wheneve f (ath) = f(a), we have Hluth) ha)-1(h) 1 IN) th) we may 2) thele we explore the relationship bln deeivative of function & decivative of its innese decvative we already For functions, whose relation to find know, we can decivatives of inners without having to Unit def of decivative. We We

In particulae, we apply formule for duivatics of inveese functions to trignomitic functions This formula may also be used to extend the power rule to rational exponents If f(x) is both invectible & differentiable, f(x) es also differentiable. consides, In the fige, if we (f-1) (a) = 2 2 2 t's" (a) = then inrees of 1 f' (f (a) where f'(f (as) - 2 Р f(n) slope t'["le)), Yer Y slope =le"/a)2 (file, a) P. crey a, there dig a function says that for that for every 2, is excatly one y. That is, y values can be duplicated but a values can't be repeated. If function has incere that is also a function, then there can only be one y for every 2. an

there is excatly one y & for tercey y, there is excally 2. one There are functions which have bocess that are not functions There are also inveers for rela hone So, for this non-lineae function, the function, in which for everye task is impossible.