Question

This OnE i Don't know Anything about it =..= i was Absent that DaY .. So HElp me With it (－－)

Answer 1

here is the Question .. let \[f(x)= x ^{3}+4x +3 \] Evaluate \[\frac{ d }{ dx} (f ^{-1} (3))\]

Answer 2

are you sure that this is the question ?

f-1(3) is a number hence constant
so the derivative of this is 0.

Answer 3

f' = 3x^2 + 4
f'(3) = 3*9 +4 = 31

Answer 4

or you need d/dx ( f-1(x)) at the point 3 ?

Answer 5

you do not need to invert the function to find the derivative of the inverse at a given number

Answer 6

one thing that should be clear from the start is that \(f(0)=3\) right? it is the constant
therefore you know \(f^{-1}(3)=0\) we need that number

Answer 7

(×_× )

Answer 8

Some oNe tell me what is the first SteP ? <>_<>

Answer 9

i wrote the first step. the first step is figuring out what \(f^{-1}(3)\) is

Answer 10

how -.-

Answer 11

you pretty much have to make a good guess for this one, but it was easy to find

Answer 12

i thought "i must find \(f^{-1}(3)\)" meaning
"i must think what i would plug in to get 3 out" and there was the constant 3 staring me in the face, so i reasoned that \(f(0)=3\) for sure

Answer 13

so now we put aside the fact that we know \(f^{-1}(3)=0\) and then see how to find the derivative of an inverse function

Answer 14

(-_-) what r u talking about

Answer 15

i was describing in the best way i know how, the method i used to find \(f^{-1}(3)\) you need that number, as you can see from the question

Answer 16

without it, we cannot start

Answer 17

OMg .. i Feel like im the dumpiest Person on earth (º_º')

Answer 18

i can write an equation if you like
we can solve
\[x^3+4x+3=3\] for \(x\) and see what we get

Answer 19

step one
\[x^3+4x=0\] step two
\[x(x^2+4)=0\] and therefore \(x=0\)

Answer 20

should i take the derivative for the function f(x) first ?

Answer 21

nope

Answer 22

(π_π)

Answer 23

lets make it clear what the steps are
1) first find \(f^{-1}(3)\)
2) then find an expression for \(\frac{d}{dx}[f^{-1}(x)]\) in terms of \(f\), \(f'\) and \(f^{-1}\)
3) then evaluate

i will write the steps and maybe it will be clear

Answer 24

step one is done, we know \(f^{-1}(3)=0\)
so far so good?

Answer 25

from where we got Zer0

Answer 26

u mean u found the value for x

Answer 27

yes

Answer 28

but there will be more than one value for x

Answer 29

i found the value of \(x\) that would make \(f(x)=3\)

Answer 30

ohh yeah ur right

Answer 31

if there is, then the function is not invertible (has no inverse) but no, there is only one value

Answer 32

yeah yeah ur right :B

Answer 33

ok so far so good then? we know \(f(0)=3\) and so \(f^{-1}(3)=0\) yes?

Answer 34

ummm okie So far so Good :D

Answer 35

ok good. now the following is always true
\[\frac{d}{dx}[f^{-1}(x)]=\frac{1}{f'(f^{-1}(x))}\]

Answer 36

noo ur wrong >:[
wait its not equal to zero =..=

Answer 37

i am positive that \(f^{-1}(3)=0\), and i know that because \(f(0)=3\)

Answer 38

what makes you think it is not zero?

Answer 39

>_ >'

Answer 40

okie whatever u say professor =.=

Answer 41

its Zero cuz it is ZerO

Answer 42

▼_▼

Answer 43

so we need
\[\frac{1}{f'(f^{-1}(3))}\] for for the answer, in other words
\[\frac{1}{f'(0)}\]

Answer 44

that is why it was important that we find \(f^{-1}(3)\) first.
without it we cannot compute this number
with it, it is easy enough, since the derivative of \(f(x)=x^3+4x+3\) is \(f'(x)=3x^2+4\) and so \(f'(0)=4\) making \(\frac{d}{dx}[f^{-1}(3)]=\frac{1}{4}\)

Answer 45

to do this problem you need to know two facts
\[f^{-1}(3)=0\] and
\[\frac{d}{dx}[f^{-1}(x)]=\frac{1}{f'(f^{-1}(x))}\]

Answer 46

>_ > im Studying ur comments Sir .. i need TiMe ●,●ﾟ

Answer 47

take the time, all the steps are there.
the only thing i did not write is why it is true that
\[\frac{d}{dx}[f^{-1}(x)]=\frac{1}{f'(f^{-1}(x))}\]
i can explain it if you like, it is a direct consequence of the chain rule, but in order to do this problem you need to use this fact (not necessarily understand where it comes from)

Answer 48

So the final Answer iS just 1/4 ? ._.

Answer 49

yes,
we found \(f^{-1}(3)=0\)
we found \(f'(x)=3x^2+4\)
we found \(f'(0)=4\) and we know
the reciprocal of \(4\) is \(\frac{1}{4}\)

Answer 50

yeah i will see how u got the answer ^_^'

(．．;) anyways .... Thanks Sir @satellite73

sankiyo very much X3

OAO

Answer 51

yw