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Suppose $f$ and $g$ are inverse functions and have second derivatives. Show that$$g^{\prime \prime}(x)=-\frac{f^{\prime \prime}(g(x))}{\left(f^{\prime}(g(x))\right)^{3}}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Campbell University

Oregon State University

Lectures

01:15

Verify that $f$ and $g$ ar…

01:14

00:48

02:14

Suppose $f^{-1}$ is the in…

05:22

Suppose that both $f$ and …

01:23

Show that $f$ and $g$ are …

So if we know who these two functions are in verses of each other first, if we go ahead and do the derivative of G FX, this should be equal to one over f prime of G FX. And this is just, um since F and G are in versus, um, you can see this comes from that equation for they give us in the book. Now, if we want to get the second derivative, well, we can just take the derivative of this again. So d by D. X on each side. So that is going to give g double prime over here and now to take the derivative of this, we can use the fact that this is really to the negative First power and we just use chain so we can go ahead, take this, move it out front. Um, and that would give us, uh, negative one and then at the prime of G FX now to the negative second power. But then we have to take the derivative of the inside function, do the chain rule, and now we'll have to apply chain rule again. So over here we can first rewrite. This is negative one over f prime of G of X squared and then the derivative of this will change rules. Let's take the derivative the outside, which would be a double prime affects. But then we have to take the drought of the inside, which is G prime of X. Now, let's compare this with what we have up here. So in the numerator, we're only supposed to have the double derivative of that. So that means we can just change g prime out for what we have right here. So we're going to replace that with one over a prime of G of X, and then this multiplies with that, and then we just end up with where it is negative. Plus, we have negative one over. If you were actually in the numerous, it's not just negative one. It would be negative f double prime of G F X, all over and then multiplying. Those gives us a cube so f prime of G f x, all cute. And that would be our second derivative. Uh, so since we proved what they want us to put our little proof box and a smiley face because we're glad we're done with it

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