Inverse Derivatives..? Please help

[puppymegan]

Hi,
If someone could explain how to do this I would greatly appreciate it.
I've tried looking at tutorials online but I'm not sure how to apply them to my own problems.

The problem is:
f(x)=sqrt(x²+5x) for x ≥ 0
g(x) = f^-1(x)
What is g'(6)?

Thanks

 

[Deleted member 4993]

Hi,
If someone could explain how to do this I would greatly appreciate it.
I've tried looking at tutorials online but I'm not sure how to apply them to my own problems.

The problem is:
f(x)=sqrt(x²+5x) for x ≥ 0
g(x) = f^-1(x)
What is g'(6)?

Thanks

Use the fact f(f^-1(x)) = x and chain rule to derive g'(x).

also read:

https://math.libretexts.org/Bookshe...vatives/3.7:_Derivatives_of_Inverse_Functions

 

[pka]

The problem is:
f(x)=sqrt(x²+5x) for x ≥ 0
g(x) = f^-1(x)
What is g'(6)?

By the nature of inverse functions we have:
\(\displaystyle f(g(x))=x\) so that \(\displaystyle f'(g(x))\cdot g'(x)=1\) or \(\displaystyle g'(x)=\dfrac{1}{f'(g(x))}\)
Now you are looking for \(\displaystyle g'(6)=\frac{1}{f'(g(6))}\)

 

[Steven G]

To continue. so you need to know what g(6) is. Now g(x) = f^-1(x), so g(6) = f^-1(6). This is asking, f(what number) = 6
That is what should x equal so that sqrt(x²+5x). Then (x²+5x) equals, then x equals ?

 

[HallsofIvy]

That is what should x equal so that sqrt(x²+5x). Then (x²+5x) equals, then x equals ?

I believe Jomo meant to write "That is, what should x equal so that sqrt(x²+ 5x)= 6"