Question

given f(x)=3x^3+1x+2 , the value of the derivative of the INVERSE of the functioin at f^(-1) at x=-2

Answer 1

ok i will show you in few steps how to solve this problem

Answer 2

whats next?

Answer 3

sorry let me do this again ok

Answer 4

kk yeah i got the inverse im stuck after it

Answer 5

wait is this x^3

Answer 6

ok there is a different method then

Answer 7

i will be right back sorry i got a phone call

Answer 8

okay

Answer 9

back, sorry for the delay

Answer 10

do you know anything about cubic formula?

Answer 11

no

Answer 12

i will do the whole answer but it will take a while typing it

Answer 13

well this is the best method to apply in order to solve this problem

Answer 14

if you dont mind, thanks:)

Answer 15

you're welcome

Answer 16

well it is simple, as it is a bit similar to quadratic equation but a bit more messy

Answer 17

let's start

Answer 18

Step one: rearrange the equation
\[y = 3x^3+x+2\]

Answer 19

\[x = 3y^3+y+2\]

Answer 20

\[ 3y^3+y+(2-x)=0\]

Answer 21

Step two: applying the cubic root formula
\[ax^3+bx^2+cx+d=0\]

Answer 22

\[x = \sqrt[3]{\frac{- b^3 }{ 27a^3 }+\frac{ bc }{ 6a^2 }-\frac{ d }{2a }+\sqrt{(\frac{- b^3 }{ 27a^3 }+\frac{ bc }{ 6a^2 }-\frac{ d }{2a })^2+(\frac{ c }{ 3a }-\frac{ b^2 }{ 9a^2 })^3}}\]

Answer 23

\[+ \sqrt[3]{\frac{- b^3 }{ 27a^3 }+\frac{ bc }{ 6a^2 }-\frac{ d }{2a }+\sqrt{(\frac{- b^3 }{ 27a^3 }+\frac{ bc }{ 6a^2 }-\frac{ d }{2a })^2+(\frac{ c }{ 3a }-\frac{ b^2 }{ 9a^2 })^3}}-\frac{ b }{ 3a }\]

Answer 24

these two posts are one equation

Answer 25

now what are the coefficients of our third order polynomial

Answer 26

\[a=3, b=0, c=1, d=2\]

Answer 27

now in the above equation all the b in it are canceled which will simplify our expression that yields to the following :

Answer 28

\[y = \sqrt[3]{\frac{ -d }{ 2a }+\sqrt{(\frac{ -d }{ 2a })^2+(\frac{ c }{ 3a })^3}}\]

Answer 29

\[+\sqrt[3]{\frac{ -d }{ 2a }+\sqrt{(\frac{ -d }{ 2a })^2+(\frac{ c }{ 3a })^3}}\]

Answer 30

again the last two posts are one equation separated by the plus sign

Answer 31

recall \[a=3, b=0, c=1, d=(2-x)\]

Answer 32

plug in those values

Answer 33

do you know how to substitute those values in the above equation

Answer 34

i will leave the rest to you, i have to fix my customers computer, then i will be back as soon as i can

Answer 35

okay thanks