The Power Rule
Lesson 5 of 13
Objective: SWBAT use the Power Rule to find the derivative function.
My students always experience a feeling of great relief during today's lesson. Instead of always using the limit method for finding the derivative, they find a shortcut that will allow them to quickly figure out the derivative. I typically preface the lesson by saying that the work they put in today will save them tons of time in the future (this helps motivate students who may be feeling a little weary after the work of the last two days).
I give students this worksheet and have them work on questions #1 and #2 with their table groups. Then we will share out with the entire class. Here are some things I am looking for as we discuss.
- Question #1 - The derivative is the slope of the tangent line at a given point. It is found by choosing an arbitrary second point and then moving that point closer to the given point by using the limit as the distance goes to zero. It can get abstract at times, but students need to remember that a derivative is a slope. This becomes our mantra for this unit and I will make them repeat "a derivative is a slope" over and over.
- Question #2 - The instantaneous slope of any point on the graph of f(x) = 4x is going to be 4, so the slope will be 4 for any x-value. Thus the derivative function is f'(x) = 4.
Now that we have reviewed a little about derivatives, I want students to see if they can figure out the Power Rule for derivatives by themselves. I have them work on questions #3-5 of this worksheet. I tell students that they can use any method to fill out these charts - they may want to use the limit method for finding the derivative or they may want to just think it through like they did for question #1. Either way, I want them to communicate their method with the others in their table group.
In my experience, students can easily reason to find the derivative of f(x) = x. Then they use the limit definition to find derivatives for f(x) = x^2 and f(x) = x^3. At this point they will notice a pattern and will fill in the rest of the chart without verifying the results. As I notice students doing this, I want them to focus on why their answers are correct and how they could justify the claims without going through the whole algebraic process.
Once students have completed questions #3-5, I will randomly select table groups to share their results with the class. I really want to make sure that we think about why this rule works. I will usually select a few students to share their reasons and justify their claims to the rest of the class.
Thinking about the definition of the derivative and considering the algebraic steps is usually very convincing to students. In the video below I outline a way to prove to your students that the Power Rule works. If this method does not come up in your class discussion, I would definitely have a conversation about it.
For the cases of multiplying by a constant, i.e. f(x) = ax^n, you could use the same argument as in the video above but just factor the constant out of every single term.
Next we fill in the chart on underneath question #5 to summarize the Power Rule. At this point, I think that it is imperative to list examples of when you can and cannot use the Power Rule. My students have a tendency to want to use it for any function that contains an exponent (such as g(x) = (2x + 4)^3, even though we did not prove that it could be done. I stress that we can only use the power rule for functions written in the form f(x) = ax^n + bx^m + cx^p +..., where each term has its own coefficient and exponent. For g(x) above, we would have to expand (2x + 4)^3 before taking the derivative.
For homework, students will work on questions #6-9 to finish up the worksheet. This will give them some more practice with the shortcut for finding the derivative and get them thinking about the equation of a tangent line.