Defining the Derivative (Topic 2.2)

Unit 2 - Day 3 and 4

​Learning Objectives​
  • Understand that the derivative is itself a function that outputs the slope of the curve at any point on the original function.

​Success Criteria
  • I can explain what a point on the derivative graph represents.

  • I can use the limit definition of a derivative to calculate the slope of a tangent line for any x-value. 

  • I can write the equation of a tangent line.

Quick Lesson Plan
Activity: The Making of a Slopes Graph



Lesson Handout

Answer Key


This is one of our absolute favorite lessons! In this lesson we move from the informal notion of a “slopes graph” to the formal idea of a derivative. Students leave class today with a solid, tangible, well-conceptualized understanding of the derivative as a function, not just as individual slopes. The artifacts from this lesson are kept on our walls throughout the year and will be referred back to in many future lessons.

Teaching Tips

At the beginning of each lesson assign each student a whole number between -12 and 12. It is important that each student has their own number (increase the range if your class is large). This number will be their x-value throughout the entire lesson. They can fill this in whenever they see x=_____ in the lesson. If your class is small, you will have to fill in some of the dots yourself so that students start to see some patterns in the “slopes graph”. This will not detract from the lesson as long as every student is calculating their own instantaneous rate of change at their own personal x-value.

We recommend pausing to debrief after question 4, making some early remarks about why the dots all form a horizontal line and then letting students continue. We suggest not using the word derivative in place of “slopes graph” until the end of the experience.

Since this lesson is rather lengthy we strongly recommend that when you get to question 5, you assign half the students to question 5a and half the students to question 6a (stronger students can be asked to tackle the derivative of the cubic function). This way they are only doing one long derivative problem instead of 2. This means you will only have half of the dots on the paper, but in an average sized class, this will still give you a good idea of the general shape of this curve. If your students are stronger you can have them do both. It is important that EVERY student copies down the graphs and the “slopes graph” for all three functions.

When you debrief, add f’(x), g’(x), and h’(x) to your dot functions that were previously just called “slopes graphs.”

After the dots/slopes are plotted, point to a specific dot and ask which student is responsible for putting up that dot. Ask the student to explain what their dot represents. Do this multiple times during the lesson and in subsequent days so that students are familiarizing themselves with the idea that the point on the derivative represents the slope of the original curve at x=___”. Insist on hearing the word “slope”, the particular x-value, and that the slope is from the original function. It is very important that students don’t use vague language like “it” or “the graph” or even “the slope”. Train your students to use precise language like “the graph of f’” or “the slope of the original curve, f(x)”.

Exam Insights

This is guaranteed AP Test content in both the multiple choice and free response sections. The time you take to lay the foundation (graphical, analytical, and verbal) for the concept of a derivative will be repaid in student retention, understanding and performance on the AP Test! As you review past AP questions, note the different representations, the different variables, and the varied notation presented to students. We can’t emphasize enough the importance of the experiences in 2.1 and 2.2.

Student Misconceptions

Students generally do not struggle with the ‘prime’ notation but have more difficulty with the Leibniz notation (dy/dx). When assigning homework and when creating assessments, be sure to vary the notation (and use “varied” independent and dependent variables!!) so students become familiar with each kind. Students tend to also struggle with the operator notation of d/dx (y) and I tell my students this simply is the verb version of the notation, meaning it’s a command to take the derivative.

When writing equations of tangent lines, continue to stress the value of point-slope form even though students tend to prefer slope-intercept form. Some students have a hard time differentiating between the slope of a tangent line and the equation of a tangent line. Make sure students understand that the derivative gives the slope of the line, but to write the equation of the tangent line they still need all the components of a linear equation that they learned in Algebra 1.

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