Question

How do you find f'(x) using the definition of a derivative `f(x) =x^2 - 1`?

Answer 1

`f'(x)=2x`

Explanation:

The definition of a derivative is:
`f'(x)=lim_(hto0)(f(x+h)-f(x))/h`

Here, `f(x)=x^2-1`

`f'(x)=lim_(hto0)((x+h)^2-1-x^2+1)/h`

`f'(x)=lim_(hto0)(x^2+2xh+h^2-x^2)/h`

`f'(x)=lim_(hto0)(2xh+h^2)/h`

`f'(x)=lim_(hto0)2x+h=2x+0=2x`

`f'(x)=2x` for `f(x)=x^2-1`