Question

How do you find the derivative of `f(x)= 9-x^2` using the limit definition at x=2?

Answer 1

`f'(2) = -4`

Explanation:

By definition `f'(x) = lim_"h->0" (f(x+h)-f(x))/h`

In our example `f(x) = 9-x^2`

Hence `f'(x) = lim_"h->0" (9-(x+h)^2-(9-x^2))/h`

`= lim_"h->0" (9-x^2-2xh-h^2-9+x^2)/h`

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

`= lim_"h->0"-2x-h = -2x`

`:. f'(2) = -2*2 = -4`