Derivative/Formal definition
From AoPSWiki
The derivative of a function is defined as the instantaneous rate of change of the function at a certain point. For a line, this is just the slope. For more complex curves, we can find the rate of change between two points on the curve easily since we can draw a line through them.
In the image above, the average rate of change between the two points is the slope of the line that goes through them: 
.
We can move the second point closer to the first one to find a more accurate value of the derivative. Thus, taking the limit as 
goes to 0 will give us the derivative of the function at 
:
If this limit exists, it is the derivative of 
at . If it does not exist, we say that is not differentiable at .
