How do you use the limit definition of the derivative to find the derivative of #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#.