Differentiability

In Calculus, we don't use the word "derive". To "derive" means to get something from something else.

Finding the derivative is done by the process of Differentiation.

DIFFERENTIATE, NOT DERIVE.

So, the derivative of a function at a point is another function, that is determined by a limit, specifically the slope of the line tangent to the graph of the function at that point.

This may not always exist.

Recall from limits that for a limit to exist, both limits from the left and right side must exist and be equal. In other words, for the limit of a function to exist at a point, the function MUST BE CONTINUOUS AT THAT POINT.

A function can be continuous at a point. However, the limit that represents the derivative might not always exist.

YOU NEED CONTINUITY TO HAVE DIFFERENTIABILITY. BUT CONTINUITY DOES NOT IMPLY DIFFERENTIABILITY.

You need to know this because if you can prove a function is discontinuous at a point, you don't have to bother checking differentiability. It won't exist.

However, once again,

IF A FUNCTION IS DIFFERENTIABLE, IT IS CONTINUOUS. BUT, IF A FUNCTION IS CONTINUOUS, THAT DOES NOT MEAN IT'S DIFFERENTIABLE. YOU WILL HAVE TO PROVE IT.

This is where we properly learn about discontinuities and cases of non-differentiability.

Studying Differentiability