Question

How do you find f'(x) using the definition of a derivative `f(x) =x^3 - 3x+5`?

Answer 1

Use limit definition of derivative to find:

`f'(x) = 3x^2-3`

Explanation:

`f'(x) = lim_(h->0) (f(x+h) - f(x))/h`

`=lim_(h->0)(((x+h)^3-3(x+h)+5)-(x^3-3x+5))/h`

`=lim_(h->0)((color(red)(cancel(color(black)(x^3)))+3hx^2+3h^2x+h^3-color(red)(cancel(color(black)(3x)))-3h+color(red)(cancel(color(black)(5))))-(color(red)(cancel(color(black)(x^3)))-color(red)(cancel(color(black)(3x)))+color(red)(cancel(color(black)(5)))))/h`

`=lim_(h->0)(3x^2+3hx+h^2-3)`

`=3x^2-3`