How do I use the limit definition of derivative to find #f'(x)# for #f(x)=x^3-2# ?

1 Answer
Andrew I.
Jun 5, 2018

#f'(x)=3x^2#

Explanation:

#f(x)=x^3-2#

The limit definition states that:

#f'(x)=lim_(h->0) (f(x+h)-f(x))/h#

In our case, this is:

#f'(x)=lim_(h->0)(((x+h)^3-2)-(x^3-2))/h#

Straight away, the #-2#s cancel with each other, we also expand the term in the brackets to get:

#=lim_(h->0)(x^3+3x^2h+3xh^2-x^3)/h#

The #x^3# cancel each other leaving us with:

#=lim_(h->0)(3x^2h+3xh^2)/h#

Cancel the #h#s on the top with the bottom to get:

#=lim_(h->0)(3x^2+3xh)#

Evaluating this limit, there is no #h# on the first term so it will stay put. As #h# on the 2nd term tends to 0, the second term will vanish completely to leave us with:

#f'(x)=3x^2#

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