We have recently learned that we can describe the instantaneous rate of change of a function \(f\) at a value \(a\) by computing \(f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h},\) provided this limit exists. When we can find \(f'(a)\), we understand that this value represents the instantaneous rate of change of the function with respect to the input variable, and also the slope of the tangent line to the curve \(y = f(x)\) at the point \((a,f(a))\).

By viewing the constant \(a\) as a variable in its own right, we will next begin thinking about how \(y = f'(x)\) is itself a function, indeed a function that is related to – or derived from – the original function \(f\). One of the next big questions is: given a function \(y = f(x)\), can we find a graph of or formula for or other information about this new function \(f'(x)\)?

The section covers the following topics: The derivative as a function. The relationship between the graph of \(f\) and the graph of \(f'\). Computing the derivative function of basic functions using the definition of the derivative.

Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

  • State the definition of the derivative function.
  • Illustrate on a graph how a function can fail to be differentiable at a point.

Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

  • Given a graph of a function \(f\), identify the graph of its derivative \(f'\) from a list, and describe aspects of the behavior of the graph of \(f'\).
  • Sketch the graph of the derivative \(f'(x)\) given the graph of \(f(x)\).
  • Given a function, derive a formula for the derivative function using the definition of the derivative.

To prepare for class

After class

  • Finish any in-class activities you might not have finished during class.
  • Do the problems on the WeBWorK assignment for this section.


Charles Fortin Avatar Charles Fortin
Gabriel Indurskis Avatar Gabriel Indurskis


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