How do you use the limit definition of the derivative to find the derivative of $f (x) = - x^{3}$ $f(x)=-x^3$ ?

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Answer 1 · 2016-07-18T16:56:00

$f'(x)=-3x^2$ , as detailed below in the Explanation.

We recall that $f'(x)=lim_(trarrx)[(f(t)-f(x))/(t-x)]$

$f(x)=-x^3 rArr f(t)=-t^3$

$:. f'(x)=lim_(trarrx)[{-t^3-(-x^3)}/(t-x)]$

$lim_(trarrx)[-{(t^3-x^3)/(t-x)}]$

$lim_(trarrx)[-{cancel(t-x)(t^2+tx+x^2)}/cancel(t-x)].......(1)$

$lim_(trarrx){-(t^2+tx+x^2)}$

$=-(x^2+x*x+x^2)$

$=-3x^2$ .

We note that at $(1)$ , cancellation by $(t-x)$ is admissible, because, as $trarrx, t!=x$ .

How do you use the limit definition of the derivative to find the derivative of f(x)=-x^3f(x)=−x3f(x)=-x^3?