Sec. 1.4 - The Derivative as a Function

Overview

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

• Do the Preview Activity for section 1.4 (on WeBWorK if required by your teacher).

• Read the Introduction of section 1.4, taking good note of the definition of the derivative as a function (Definition 1.4.2).

• Play around with the two apps mentioned in the text: http://gvsu.edu/s/5C and http://gvsu.edu/s/5D

• Watch the following video which discusses the definition of the derivative as a function.

• Watch this video for an example of how to algebraically find the formula for the derivative function:

• Watch this video which shows how to construct the graph of the derivative (using a Geogebra app), given the graph of the function (video created at CCSL):

• Do some experimentation with the following interactive applets:

  • The Derivative as a Function

  • Try to Graph the Derivative Function

  • Identify the Derivative Function

  • Derivatives and Graph Transformations

  • Identify an Antiderivative Function (Note: An antiderivative of a given function is exactly what it sounds like: a function such that its derivative is the given function. So in this problem, you are going in the opposite direction than what you did in the previous applet.)

• Watch the following video for another example:

After class

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

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