Question

Question #0321b

Answer 1

See explanation.

Explanation:

a. Calculate the inverse function.

`y=x^3-18`

`y+18=x^3`

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

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

Answer 2

`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.!