Derivatives as functions

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After completing this section, students should be able to do the following.

Understand the derivative as a function related to the original definition of a function.
Find the derivative function using the limit definition.
Relate the derivative function to the derivative at a point.
Explain the relationship between differentiability and continuity.
Relate the graph of the function to the graph of its derivative.
Determine whether a piecewise function is differentiable.