For the function f: R ->R,f(x) = 7 - x^(2) , find: f' (-1).Show all working

Question

For the function f: R ->R,f(x) = 7 -
`x^(2)` , find:
f' (-1).Show all working

Answer 1

`f'(-1) = \pm 2√(2)`

Explanation:

For
`f: \mathbb{R} \rightarrow \mathbb{R}` such that
`f(x) = 7-x^2`, find
`f'(-1)`

Recall that

`f'(y)=x \text{ whenever } f(x) = y, \forall y \in \mathbb{R}`

Therefore,

`f(x) = 7-x^2 \implies y = 7-x^2 \implies x = 7-y^2 \implies y = \pm √( 7-x)`

`\therefore f'(x) = \pm √( 7-x)`

`f'(-1) = \pm √( 7-(-1)) = \pm √(8) = \pm √(2^3 ) = \pm 2√(2)`