Now that we have some counting from Counting, Permutations and Combinations, the Binomial Theorem, and have seen what a limit and derivative are in Dividing by zero: Imploding the Universe., we are now ready to look at the power rule for derivatives. We will therefore begin by going through an example, followed by a proof that the rule we come up with will work in general.
In a calculus class it is often helpful not to dive directly into a proof of something. Instead, I tend to ease into these proofs by working on an example or two so that we can outline the process. To this end, I would go through finding the derivatives of a constant k, and the function x and x2 with the students. The nice things about these examples is that we can check to see if we are correct, because the functions themselves are the tangent lines, and finding the slope is something that has been done before.
I won’t run through these here, however, I do want to show how I would find the derivative of f(x)=x3 with my students. To do this, I would remind them that
I’m using the same definition we found in Dividing by zero: Imploding the Universe., however, I have substituted an h for Δx since it is a little easier to write. In order to calculate the derivative of f(x)=x3, I break the process up into four smaller steps. I first learned this process from Dr. Karl Byleen when teaching a business calculus course with his book Calculus for Business. The process really just explicitly states what you would naturally do when trying to complete a complicated calculation. However, by walking your students through this and making them follow the steps, there are significantly fewer mistakes made on homework and tests. The process is,
- First find f(x+h) and simplify. Note that instead of multiplying out (x+h)3, I would instead remind them of the binomial theorem because we will be using this for general case.
- We would then find and simplify f(x+h)-f(x).
- Then, find and simplify the entire inside.
- Finally, find the limit noting that in the limit h is not 0, so we can substitute what we found.
We have now shown that that derivative of x3 is 3x2. Even at this point, when combined with the fact that the derivative of x2 is 2x, the students start to see the pattern. If no one can see it, I will show the derivative x4 is 4x3. Otherwise, I tell my students that I’m happy they’ve seen the pattern.
However, I also take this time to point out that we’ve applied argument due to inductive reasoning. That is, we saw some examples and tried to expand this. Is this guaranteed to work? If they say, yes, I will provide examples of misleading patterns, for example you’re walking and you go up steps. You’ve moved up 1, 2, 3, 4 steps, what will happen next? Whatever they choose, pick something different. That is, if they say you will have moved up 5 steps, state the stair case stopped and you stay at height 4. If they say 4, say it kept going up. If they say both, say the staircase ended and you fell back to level 0.
The goal here is to show that while we’ve made an educated guess, the pattern may change for some unseen reason. Therefore, we need to provide a deductive argument to show that the pattern will always hold. I would then continue to give the proof of the power rule.
We are now ready to show the power rule for f(x)=xn for all natural numbers n>1. I would now point out that our only option is to use the limit definition of derivative.
- We would first use the binomial theorem to findIf you hadn’t previously explained the binomial theorem, make sure to let your students know what you mean by this notation. In particular, explain why the first two coefficients are 1 and n, and explain that the rest are just some number.
- We then subtract f(x) and getNote that since the sum ends at i=n-2, there is still at least one h in each of the terms of the sum.
- If we now divide by h, we get
- Finally, taking the limit, we find Seeing that each term in the sum goes to 0 depends on seeing that there are still hs in each term, and you therefore get a number multiplied by 0.
We have now proven that the power rule that they saw earlier,
is true for all n>1. That is, not only do we think it will work because we saw a pattern, but we know that it will always work. I really try to emphasize this to my students so that they will understand the distinction between the guessing and knowing. I want them to realize that the equations and properties they’ve been using in mathematics aren’t true just because someone claimed them to be true, but rather someone took the time to prove, based on supplied assumptions, that the formulas used are correct.
I also point out, that our proof doesn’t work for n=0,1, but we can work with these situations with a few adjustments. In particular, the above formula would give that the derivative of x is, 1x1-1=x0. However, the derivative of x is 1. While these agree for all x except 0, we do note that 00 is undefined. We have a similar situation if we write 1=x0. Then the derivative is 0x-1. Again, there is a distinction between these at x=0, so I state they can use the power rule for these cases with the caveat that they need to be careful at this one point.
The importance of understanding why something is true and that we don’t just make claims without justification in mathematics is something that I really try to get across to my Calculus students. I feel like this is normally the first time students are introduced to underlying principles of mathematics. I hope that I can get a few of them to understand what mathematics will become if they continue with their courses. Realizing that mathematics isn’t just a box of tools that you pull out, either excites or terrifies students. Either way, it is good for them to know at this point instead of later in their studies.
Next time in this series, we will continue to work with the power rule. While it may seem we have finished, we have only shown the power rule works for all natural numbers. However, we will also need to look at the case where the powers are negative or rational.
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