Question

compare the right-hand and left-hand derivatives to show that the function is not differentiable at the point P. find all points where f is not differentiable. (graph attached)

Answer 1

apparently the answer is
left hand derivative: 1/2
right hand derivative: 2
but that makes no sense to me at all

Answer 2

yes... it should not make sense.
That is because the question asks to "show that the function is NOT differentiable at point P."
Since the answer you've got is nonsense because the left hand derivative does not match the right hand derivative, you have just proven that this function is NOT differentialble at point P.