Question

Calculus - Inverse of a derivative-using table values

Answer 1

Find the value of \[\left[ f ^{^{-1}}(2) \right]'\]

Answer 2

I got 3 but I am not sure that is the correct solution

Answer 3

@ganeshie8

Answer 4

lets do a quick+easy derivation

Answer 5

we start with :
\[f^{-1}(f(x)) = x\]

Answer 6

differentiate both sides using chain rule :

\[\left(f^{-1}(f(x))\right)' = \left(x\right)'\]

Answer 7

remember that is how we test if functions are truly inverses

Answer 8

so ganeshie8 is correct

Answer 9

Yeah, I got it haha.

Answer 10

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

Answer 11

\[\large [ f^{-1}(f(x))]' = \dfrac{1}{f'(x)}\]

Answer 12

see if that looks okay.. @precal @iambatman

Answer 13

actually there is a mistake..

Answer 14

\[\large [ f^{-1}(f(x))]' = \dfrac{1}{f'(x)}\]

when f(x) = 2, x = 1

\[\large [ f^{-1}(2)]' = \dfrac{1}{f'(1)} = \dfrac{1}{3}\]

Answer 15

that should work ^ please check...

Answer 16

ok I like the way you explained it. I was always given the formula but I had no idea how it was derived

Answer 17

Yeah that seems right, I was just looking at where you got you got x = 1, but it's from the chart haha.

Answer 18

yes :)

Answer 19

ok lets see if I did the second one correct\[\left[ g ^{-1} (1)\right]'\]

Answer 20

is it -2

Answer 21

can you check that for me @Iambatman

Answer 22

@iambatman

Answer 23

sorry computer capitalized the i

Answer 24

@ganeshie8
Will you please help me with another question? It is related to the same table?

Answer 25

sure, ill try.. btw -2 is right..

Answer 26

Find the equation of the line tangent to the graph of g' when x=-2 if g'(-2)=2/3
Thanks, you were a great teacher that is why I got it correct

Answer 27

I know the slope I am to use is 2/3 but I am not sure what point I am to use since x=-2 is not on the table

Answer 28

I know the given slope tells me that the function is increasing because it is positive but I do not know what to use

Answer 29

do I use (1,-2)?

Answer 30

y+2=(2/3)(x-1)

Answer 31

g' is the original function,
(-2, 2/3) is a point

we need to find the slope g''(-2)

Answer 32

... slope of g' is g''

Answer 33

How am I going to do that?

Answer 34

is that all the information given ?

Answer 35

I guess I am still assuming g is the original function of g' since I was given the table but you probably are on the correct path

Answer 36

yes with the same table

Answer 37

not sure how we can compute g''(-2) from the table :/

Answer 38

@ikram002p wanna try

Answer 39

Is the table the only thing you're given?

Answer 40

yes

Answer 41

Here is the complete question :
`Find the equation of the line tangent to the graph of g' when x=-2 if g'(-2)=2/3`

use below table :
http://assets.openstudy.com/updates/attachments/53aec82fe4b02c45b015a24c-precal-1403963562053-doc2.pdf

Answer 42

and I am told g'(-2) =2/3

Answer 43

sure

Answer 44

Lol is there a graph?

Answer 45

(-2, 2/3) is a point on g'
so the tangent line will be :

y - 2/3 = m(x+2)

we need to find m

Answer 46

m = g''(-2)

Answer 47

no, sorry just table of data given. This is AP calculus btw so they tend to try to confuse you

Answer 48

I thought 2/3 is the m value

Answer 49

2/3 is the slope of the tangent line

Answer 50

you're right

Answer 51

but I am missing the point

Answer 52

slope of the tangent line = derivative

Answer 53

if g' is linear then we might have two point and find the slope ?

Answer 54

slope at g(a) is g'(a)
slope at g'(a) is g''(a)

Answer 55

careful, we're trying to find the tangent line of the graph g', NOT g

Answer 56

Bleh right

Answer 57

can't we just use 2 points to calculate that slope?

Answer 58

yeah but u have
(-3 ,-1/2)
(1 ,-2)
(2, -6)

Answer 59

g'(1) and g'(-3) since -2 is between

Answer 60

but mm i cant assume that is correct unless g' is linear

Answer 61

that gives you slope of secant line, but the question is asking us to find the slope of tangent line

Answer 62

then lets try , definition of tangent / derivative ?