Question

Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function:

f(x) = x2 + 1 for x is greater than or equal to 0. (5, 2)

Answer 1

So we need to find the derivative of the inverse.
That means we first need to find the inverse of the function that if y=x^2+1 ,x>=0

Answer 2

do you know how to solve y=x^2+1 where x>=0 for
x?

Answer 3

f^-1 = sqrt(x - 1)

Answer 4

if you don't want to do this way
there is actually another way

Answer 5

I will show you when we are done

Answer 6

ok and then find the derivative of that by using chaining rule

Answer 7

chain rule*

Answer 8

if you don't know chain rule yet we should do it the other way

Answer 9

\[\frac{d}{dx} f^{-1}(x)=\frac{1}{f'(f^{-1}(x))}\]

Answer 10

\[\frac{d}{dx} f^{-1}(x)|_{x=5}=\frac{1}{f'(f^{-1}(5))}\]

Answer 11

f inverse of 5 is given

Answer 12

by that ordered pair (5,2)

Answer 13

so 2?

Answer 14

\[\frac{d}{dx} f^{-1}(x)|_{x=5}=\frac{1}{f'(f^{-1}(5))} =\frac{1}{f'(2)}\]

Answer 15

so instead of finding the derivative of f inverse
we could find the derivative of f

Answer 16

but either way will work for this one

Answer 17

i will show you the other way when we are done with this way

Answer 18

derivative of 2 is 0.

Answer 19

no no ...

Answer 20

to find f'(2)
you need to first find the function that is f'

Answer 21

so f'(x)=?

Answer 22

x^2 + 1
so derivative is x

Answer 23

2x right?

Answer 24

2x, yup

Answer 25

and since f'(x)=2x
then f'(2)=2(2)=?

Answer 26

4

Answer 27

\[\frac{d}{dx} f^{-1}(x)|_{x=5}=\frac{1}{f'(f^{-1}(5))} =\frac{1}{f'(2)} =\frac{1}{4}\]

Answer 28

ok we can do it the other way too

Answer 29

\[f^{-1}(x)=\sqrt{x-1}\]

Answer 30

\[\frac{d}{dx}(f^{-1}(x))=\frac{1}{2\sqrt{x-1}}\]

Answer 31

so\[\frac{d}{dx}(f^{-1}(x))|_{x=5}=\frac{1}{2\sqrt{x-1}} |_{x=5}=\frac{1}{2\sqrt{5-1}}=\frac{1}{2\sqrt{4}}=\frac{1}{2(2)}=\frac{1}{4}\]