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Answer 1 · 2018-01-27T19:57:20

#(f(x+h)-f(x))/h=3x^2+3xh+h^2-1#

We want to find #(f(x+h)-f(x))/h# for #f(x)=x^3-x#:

#f(x+h) = (x+h)^3-(x+h)#

#= x^3 +3x^2h+3xh^2+h^3-x-h#,

so substituting gives:

#(x^3 +3x^2h+3xh^2+h^3-x-h-(x^3-x))/h#

#(x^3 +3x^2h+3xh^2+h^3-x-h-x^3+x)/h#

#(3x^2h+3xh^2+h^3-h)/h#

#3x^2+3xh+h^2-1#

So

#(f(x+h)-f(x))/h=3x^2+3xh+h^2-1#