Derivative of an inverse function

[thegoosegirl42]

Homework Statement

I will post a picture of the problem and then the second picture will be my work. The problems are the first two.

Homework Equations

The Attempt at a Solution

I didn't know how to do this at first so I don't know if I am doing it correctly now. Also I don't know the correct answer.

 

[WWGD]

Try using $f(f^{-1}(x))=x$ and differentiate both sides. EDIT I am assuming the necessary conditions hold.

 

[vela]

Please don't post images in lieu of typing out the problem statement and work.

The problems are to find (a) $(f^{-1})'(13)$ given that $f(x)=2x^3-7x-20$, and (b) $(f^{-1})'(6.5)$ given that $f(x) = \frac{2}{x^2}+3x$. (Would it have killed you to type that out?)

Your approach isn't correct. For one thing, you're writing upside down. What's more important, you're finding $(f')^{-1}$, but you're asked to find $(f^{-1})'$ evaluated at the given point.

Use WWGD's hint (or look in your textbook or notes) to find the relationship between $f'(x)$ and $(f^{-1})'(y)$ where $y=f(x)$. You could also think about how the slope of the curve at a point changes when you reflect the curve about the line y=x.

 

[Mark44]

For one thing, you're writing upside down.

She might be writing the right way up, but the posted image of her work is upside-down.
@thegoosegirl42, if you post images, please take care to post them with the right orientation. Many helpers won't bother with posts where the work is shown in an image, let alone where the image is sideways or upside-down.

 

[thegoosegirl42]

She might be writing the right way up, but the posted image of her work is upside-down.
@thegoosegirl42, if you post images, please take care to post them with the right orientation. Many helpers won't bother with posts where the work is shown in an image, let alone where the image is sideways or upside-down.

Ya, I didn't check to make sure if it was oriented correctly. I just assumed it was.

 

[thegoosegirl42]

Please don't post images in lieu of typing out the problem statement and work.

The problems are to find (a) $(f^{-1})'(13)$ given that $f(x)=2x^3-7x-20$, and (b) $(f^{-1})'(6.5)$ given that $f(x) = \frac{2}{x^2}+3x$. (Would it have killed you to type that out?)

Your approach isn't correct. For one thing, you're writing upside down. What's more important, you're finding $(f')^{-1}$, but you're asked to find $(f^{-1})'$ evaluated at the given point.

Use WWGD's hint (or look in your textbook or notes) to find the relationship between $f'(x)$ and $(f^{-1})'(y)$ where $y=f(x)$. You could also think about how the slope of the curve at a point changes when you reflect the curve about the line y=x.

I figured posting a picture would be easier for people to see than typing it out. I for one don't like these problems typed out.

 

[Mark44]

WWGD and vela used LaTeX to format what they wrote, and this isn't actually very difficult to do. Here's a link to a help page on how to do this: https://www.physicsforums.com/help/latexhelp/

You can also use BBCodes to do things like write exponents and less complicated expressions. Here's a link to a help page for that: https://www.physicsforums.com/help/bb-codes

 

[Mark44]

I figured posting a picture would be easier for people to see than typing it out. I for one don't like these problems typed out.

It's actually easier for us if you type the problems. That way we can insert a comment at the exact place where something is wrong. With an image, we can't do that, and we have to describe where the error is.