Derivative
The first derivative of a function may be calculated as
Note that in order for the limit to exist, both lim h→0+ and lim h→0− must exist and be equal, so the function must be continuous. However, continuity is a necessary but not sufficient condition for differentiability.
Explanation
The derivative of a function f with respect to the variable x is defined as
but may also be calculated more symmetrically as
provided the derivative is known to exist.
Example 1
You can see that the first derivative for the function
at the point x = 3 gives