Transition from thinking of the derivative at a point to thinking of the derivative as a function.

• Visualize the graph of a function’s derivative by considering the slope of the graph of the original function
• Relate increasing/decreasing behavior of the function to the sign of its derivative

• slope of a tangent line
• derivative at a point
• derivative function

This lesson involves making the transition from thinking of the derivative at a point (i.e., as a numerical value associated with the local slope at a particular location on the graph of a function) to thinking of the derivative as a function (by considering the numerical calculation as a process that can be employed across a domain). Students will use two familiar function examples (y = f(x) = x² and y = f(x) = sin(x)).