How do I obtain the derivative of the function by using the Inverse Function Theorem?

[reef marlin]

I eventually get to this step (last image), but how do I proceed from here?

[opal moth]

Let $f(x)$ be some differentiable function with an inverse. Let $g(x)=f^{-1}(x)$ be the inverse function. Then $g'(x)=\frac{1}{f'(f^{-1}(x))}$

[weary estuary]

SWR

[reef marlin]

I tried that, as shown in the image. But I get stuck.

[opal moth]

There is no image of your work

[reef marlin]

No? It’s the last image

[opal moth]

Oh I see

Discord preview is gaslighting me

only one image was here

you have it correct up to here

[reef marlin]

Oh ok. Where do I proceed from here?

[opal moth]

solve for g'(x^5)

then make the sub x^5 -> x

[reef marlin]

How? Can you show me?

[opal moth]

solve for it here

[reef marlin]

How? If my goal is to get the g’

[opal moth]

To be more clear, solve for g'(x^5) here

[reef marlin]

Yes. I can’t figure out how

[opal moth]

Would you know how to solve for $b$ in $a=\frac{1}{b}$?

[weary estuary]

SWR

[reef marlin]

g’(x^5)=(1/(5x^4))

[reef marlin]

But what do I do to find g’(x^5) at this point?

[opal moth]

It's a little awkward because you have g'(x^5) and you want g'(x)

So, instead write $g'(u^5)=\frac{1}{5u^4}$ and let $x=u^5$ to find $g'(x)$