Question

How do you use the definition of the derivative?

Answer 1

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!