Question

How can i describe the x-values at which f is differentiable at `f(x)=2/(x-3)` ? what is differentiable anyway ?

Answer 1

Please see below.

Explanation:

Briefly, "differentiable" means "can be differentiated" and that mean "has a derivative".

The verb, "differentiate" means "find the derivative".
The noun "differentiation" is the act or process of differentiating, hence, the act or process of finding a derivative.

Definition

Function `f` is differentiable at `x=a` if and only if `f'(a)` exists.

For this function

`f(x) = 2/(x-3) = 2(x-3)^-1 = -1(x-3)^-2 = (-2)/(x-3)^2`

`f'(x)` sn defined for all `x` except `x=3`. So `f` is differentiable at every `x` except `x=3`.