# Derivatives as functions modules

## Overview

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.

# Wait for the right moment

## Ximera Module

Two young mathematicians discuss derivatives as functions.

# The derivative as a function

## Ximera Module

Here we study the derivative of a function, as a function, in its own right.

# Differentiability implies continuity

## Ximera Module

We see that if a function is differentiable at a point, then it must be continuous at that point.