Question #0321b

2 Answers
Feb 6, 2018

See explanation.

Explanation:

a. Calculate the inverse function.

y=x^3-18y=x318

y+18=x^3y+18=x3

x=root(3)(y+18)x=3y+18

So the inverse function is: g(x)=root(3)(x+18)g(x)=3x+18

b. Calculate the derrivative

g'(x)=1/(3*root(3)((x+18)^2))

c. Substitute x=9

g'(9)=1/(3root(3)((9+18)^2))=1/(3root(3)((27)^2))

g'(9)=1/(3root(3)(729))=1/(3*9)=1/27

Feb 6, 2018

1/27.

Explanation:

Since, g(x)=f^-1(x), g(f(x))=x.

Differentiating w.r.t. x, using the Chain Rule, we have,

g'(f(x))*f'(x)=1................(star).

For g'(9), we must have, here, f(x)=9, i.e., x^3-18=9.

:. x=3.

Sub.ing x=3 in (star), we have,

g'(f(3))*f'(3)=1...............(starstar).

But,

f(x)=x^3-18 rArr f'(x)=3x^2 rArr f'(3)=27, and f(3)=9.

Utilising these in (starstar), we finally have,

g'(9)=1/(f'(3))=1/27.

Enjoy Maths.!