a-suppose-f-is-a-one-to-one-differentiable-function-and-its-inverse-function-f-1-is-also-differentia

Question

(a) Suppose $ f $ is a one-to-one differentiable function and its inverse function $ f^{-1} $ is also differentiable. Use implicit differentiation to show that
$ (f^{-1})'(x) = \frac {1}{f'(f^{-1}(x))} $
provided that the denominator is not 0.
(b) If $ f(4) = 5 $ and $ f'(4) = \frac {2}{3}, $ find $ (f^{-1})'(5). $

Frank Lin · Accepted Answer

So it arrived This implicit differentiations depreciation or sometimes going Rhys function zero. So if if. Why? Because have inverse off X, the emcee for self off. Why, then you take the RHIB here who both sigh with respect to X, we get private. Why times y prime And therefore And now we know wise If you were so pecks so we have one because if prime of f universe x times have reverse off x right, therefore that leads to the after into a division that is to the first former So what is f inverse prime of five? Keep another given all those information Well, if by the former all off five equals of prime off f inverse who five No one but everyone&#x27;s so fires for so have prime for and have prime force to over three So university we guess three over too

Amy Jiang · Answer

Okay. So as to find the following we were called out from their seven we have that&#x27;s equal to the derivative of our function and then are in versed. ATS are points five. Okay, so we know we&#x27;re giving that four is equal to five. So that means inverse of five is equal to four. That gives us one over that crime at four. How did I notice? Well, I know that our function is, um it has an inverse function, so that means our domain of our function is equal. Could range of our in verse on their veins of our function is equal to domain of our Okay. I know f prime at four. That people through to over three. So we have one over to over three and that she put trio

Robert Daugherty · Answer

section 3.8 Problem number 29. They give us a function f of X coastline expose three x show that this function has a different chewable inverse. Okay, so we were given a theorem here. Ah, in your book, I think it was the&#x27;re, um three. And if if f is defensible, So we had the original function. If it&#x27;s defensible on an interval, then and we know that if F prime of X does not equal zero on that interval, then that&#x27;s a basically a way of telling us that we know if the if the derivative is not zero, it cannot be 1 to 1. Then we know that f inverse exist and is differential. Okay, so all I really need to do here is find the derivative and show that it&#x27;s never zero. Okay, so f prime of X is going to be equal to minus sign of X plus three. Okay, let&#x27;s try setting this equal to zero. So if a negative sign of X plus three is equal to zero, that means that the sign of X is equal to three. Well, this cannot happen. We know that the sign is a number between negative one and one. So this has no solution. So because F prime is never zero, then I know that the inverse exist and that in verse is defensible according to the theory. So again, all I did was find the derivative showed the derivative cannot be zero. So if the derivative cannot be zero, then I&#x27;m never gonna have a scenario like this because that&#x27;s where the river we zero. So I&#x27;m never gonna have a function that is not one toe one. Um, so I know that it will have an inverse, and this inverse will be differential. Now we&#x27;re asked if I ng f zero and f prime of zero So again, what was our definition? F of axe? So at f of X is equal to co sign X plus, was it three x co sign X plus three X and then we had that f prime of axe is equal to minus sign of X plus three. So f of zero is going to be one plus zero, which is one and then f prime of zero is going to be zero plus three, which is three now in part. See who has a couple things find the inverse so f inverse of one. So what do we know? We know that f of zero is one. So from from part B, we found out that f of zero is equal to one. So then, by definition, by definition, off f inverse, we know that f inverse of one is equal to zero. Now, a little bit more work we&#x27;re gonna have to do for the next part. We&#x27;re gonna have to find f in verse, so f in verse, prime of one. So to find FM verse prime of one. So let&#x27;s start with what we know. We know that why is equal to co sign of X plus three x So that&#x27;s what we know about the function now to find the inverse. What do you do with in verses? You switch X and Y So let&#x27;s switch X and why? To get the behavior of the inverse. So the inverse is going to be ex equal co sign of why plus three why? And so now let&#x27;s differentiate this with respect to X. So when I differentiate this so what I&#x27;m gonna do with this is use implicit differentiation to find the derivative with respect to X and I get one is equal to So that&#x27;s gonna be minus sign of Why do I d x plus three? Do you? Why? Dx So this tells me that D y d x times three minus sign A Why is equal to one? So do I. D x is equal to one over three minus. Sign off. Why? And so then what am I trying to figure out here? Um, trying to figure out then if I&#x27;m looking at, um So we&#x27;re down to almost there. We&#x27;ve got d Y d X is equal to one over three minus sign. Why one over three? Minus a sign of why now what we know already. Um, we know that, um, if we look at the derivative, um, we know that f prime. So in this case, we have d i. D. X is one over three minus sign. Why? So we know that when y is equal to zero, this just simply evaluates to 1/3. Okay, so we find that f in verse prime of one is equal to 1/3

Amy Jiang · Answer

Okay, so let&#x27;s start by finding for us to find you prime at two. So that&#x27;s ever finding, um, are derivative of G. So we can use our Kocian role. That&#x27;s preservative of one, which is they&#x27;re all and then minus distributive of this time at X and in times one over f inverse, that&#x27;s X quiet. Okay, so we know that f evaluate at areas you could too. So since our function is, um, it has an inverse function. We also know that given this we have a inverse at two. What s equal to three. Well, that&#x27;s what we need to find you prime at two. What? I see four to negative F in verse. Derivative at you over f been Ruth at two. Glad. Okay. So the derivative of our inverse function, we know that Well, that is equal to Hi there. I&#x27;m seven. We have what, over half prime at F in verse at two. This is all over a crime, or it&#x27;s our f in verse right to quit. So let me simplify it out further. So we get negative one over f in this at two when an f crime and converse at two Okay, so we know what? Out of interest. Too difficult to three. So we get three squared in an F crime at three. So f prime at three deaths. With you one overnight, we get negative one over and nine times one overnight, and that&#x27;s this equal to negative one.

Doruk Isik · Answer

in this problem, there is a function, and we know that that function has a story in that one Warner function processor in the world, given day, everything on your the diameter att, that exact point. So we don&#x27;t have a function passes through this point for when exes, too sanctions those or also meet our narrative emulates that function. Uh, s a group one over three. And we arrested Flying the damaging off dysfunction humors humors X it For now, we know that David all functions worse. It will let that one point in this case that boy is for we&#x27;ll be equal war over functions, narrative able like another. Since we know that dysfunction passes through this point emotional to past that point for on this will be equal to a bunch of their with their privilege. Uh, this point, since this is given right here, and charitable be won overwhelming feet and derivative off its fevers. Evelyn, that, uh, this point will be equal to three

Linh Vu · Answer

No question, we are given a function. Why in could you f ax and then Ana FX has inverse function and it means then they exist half invisible X and then the born Jew far. And he&#x27;s on, uh, graph off F X and now and upon Jew far and has does laugh off Would you want out of tree here? And we know the slot on the function exam ico Judah m Prime and that value X hair. So we could, you know, am prime up to and from hand a question asked her to find ah, the off women voters I have thanks over the ANC&#x27;s here and a very off X. Could you far so by the formula fund of the river to under invest function We know that the d the f off the if thanks. I&#x27;m with the X we ico Juwan over de agua de anc&#x27;s I&#x27;m the f of X and therefore we can get this one. We go to one dividing by and, uh, and Bram off the to because hand under June and why it could you fall and we should get echo Jew. One given me one under a tree and a good your three, and this would be the answer

Frank Lin · Answer

in this public park every 1st 1 show that at the worst time back to be won over 100 years off. Ex found me know that off. Been worse off eggs. It&#x27;s a function is one tour is equal X. He detected evidence off both sides with respect to X. You see that F prime off in search of X times. They&#x27;re a joke in your function. Clams F in worse prime of X. Is it one from this three diesel issue that f commerce prime of eggs is equal to than one over f crime off a humorous off X. And that is same as what is given in the problem statement. All right, that was part of it in part B. Where has to find the worst time? Five. So we&#x27;re gonna use the question that we just drive some F universe Primer five is equal to one over. Prime asked in Worse off fire. What? Has that been worse? A fire. Well, you know that effort for is it for if 405 it means that you&#x27;re worse off 54 So then we have on over prior or four and primal forces, Given that history, he said. It is over to over three, so we find the answer to be then three or two.

Problem

(a) Show that $ f(x) = x + r^x $ is one-to-one. …

Problem 77 Hard Difficulty

Answer

(a) $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$
(b) $$ \frac{3}{2} $$

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(a) $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$(b) $$ \frac{3}{2} $$

Calculus

Anna Marie V.

Heather Z.

Caleb E.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartCalculus

Anna Marie V.

Heather Z.

Caleb E.

Joseph L.

(a) $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$
(b) $$ \frac{3}{2} $$

James Stewart

Calculus