We use the limit definition, putting in x instead of a:
Therefore f'(x) = 2x. If we plug in 1 for x we find f'(1) = 2, which agrees with our earlier calculation.
In formulas, the limit definition of the derivative function is
This is the same as the limit definition of the derivative of a point, but with x instead of a. When we evaluate the derivative of f at a point, we take a value and plug it in for x in the definition above.