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

1 Answer
George C.
Nov 1, 2015

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#

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