Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=x^2`?

Answer 1

`f'(x)=2x`

Explanation:

Given a function `f(x)` then by definition `f'(x) = lim_"h->0" (f(x+h)-f(x))/h`

In our example `f(x) = x^2`

`:. f'(x) = lim_"h->0" ((x+h)^2-x^2)/h`

`= lim_"h->0" (x^2+2xh+h^2 - x^2)/h`

`= lim_"h->0" (2xh+h^2 )/h`

`= lim_"h->0" (2x+h )`

`=2x`