Show-that-the-function-f-x-x-3-is-derivable-at-all-points-x-0-R-and-that-f-x-0-3x-0-2-

Question Number 95643 by Ar Brandon last updated on 26/May/20

Show that the function f (x) = x^{3} is derivable at all

points x_{0} \in R and that f^{'} (x_{0}) = {3 x}_{0}^{2}

Answered by Rio Michael last updated on 26/May/20

lets choose a general point c \in R,

if f (x) = x^{3} is derivable or differentiable at x = x_{0} = c then

\lim_{x \to c} \frac{f (x) - f (c)}{x - c} must exist .

but f (x) = x^{3} \Rightarrow f (c) = c^{3}

\lim_{x \to c} \frac{x^{3} - c^{3}}{x - c} = \lim_{x \to c} \frac{(x - c) (x^{2} + 2 c x + c^{3})}{(x - c)} = \lim_{x \to c} (x^{2} + 2 c x + c^{2}) = 3 c^{2}