Showing differentiability of a function at a point

Question

I am trying to understand a concept, so I will use an example. The following function is given:

[\left { \begin{matrix} 3x^{2} &x\leq 1 \\ ax+b & x>1 \end{matrix} \right.]

I wish to find for which values of a and b, it is differentiable at x=1.

I was taught that the way to do it, is to show that it is continuous, i.e., to compare the limits of the function from right and left of 1, and then since continuity is not sufficient for differentiability, compare the limits from the right and left of the derivative. I do not understand the rationale behind this. Why comparing the limits of the derivative is sufficient to claim differentiability ?

Answer 1

A continuous function is differentiable at a point iff the right hand derivative is equal to the left hand derivative. So it is not the limit of the derivatives you have to consider but the right hand derivative and the left hand derivative (which are $6$ and $a$ in this case).