How do you use the definition of the derivative?

1 Answer
Apr 19, 2017

The definition of the derivative may be of help. For instance:

Prove that the derivative of #f(x) = x^2 - 1# is #f'(x) = 2x#

By the definition of the derivative.

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

#f'(x) = lim_(h->0) ((x + h)^2 - 1 -(x^2 - 1))/h#

#f'(x) = lim_(h->0) (x^2 + 2xh + h^2 - 1 - x^2 + 1)/h#

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

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

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

#f'(x) = 2x#

Which is what we needed to show.

Practice exercises

#1#. Show that the derivative of #g(x) = (x + 2)^2# is #g'(x) = 2x + 4#.

Hopefully this helps, and good luck!