Section 4.2 Warning: Be Sure You Understand the Purpose of This Chapter
You’ve got to learn your instrument. Then, you practice, practice, practice. And then, when you finally get up there on the bandstand, forget all that and just wail.―Charlie “Bird” Parker, 1937—195512
https://en.wikipedia.org/wiki/Charlie_Parker
Imagine a surgeon using a scalpel, an auto mechanic using a torque wrench, or a professional golfer swinging a club. These people endure countless hours of practice learning to use the fundamental tools of their trade. It is not fun, nor do they expect it to be. They do it in order to become so skillful that they can use their tools seamlessly; so they can focus on what they are doing, not how they are doing it. They don’t think of the tool, they think of the task. The tool simply becomes an extension of their hands. In precisely the same way, the differentiation rules we’ll be discussing in this chapter are the basic tools of your trade and you need to become so skillful with them that you can use them without thinking about them.
| The Constant Rule | If \(a\) is a constant then \(\dx{a}=0\) |
| The Sum Rule | \(\dx(x+y)=\dx{x}+\dx{y}\) |
|
The Constant Multiple Rule |
If \(a\) is a constant then \(\dx(ax)=a\dx{x}\) |
| The Product Rule | \(\dx(xy)=x\dx{y}+y\dx{x}\) |
| The Power Rule | If \(n\) is a rational number then \(\dx(x^n)=n x^{n-1}\dx{x}\) |
| The Quotient Rule | \(\dx\left(\frac{x}{y}\right)=\frac{y\dx{x} -x\dx{y} }{y^2}\) |
The computational rules in the table above are the basic tools of Differential Calculus. They need to become an extension of your mind. You need to be able to perform these calculations while holding a larger problem in your mind. Sometimes a much larger problem. You need to become so familiar with them that they become the simple part of bigger problems.
The purpose of this chapter is to give you practice with these computations; to give you a chance to internalize them before you have to use them in a larger context. For that reason we will be providing a lot of drill problems for you to practice on. Do them.
These drill problems are not important in themselves, but once we move past this chapter it will be assumed that you can compute a differential easily — even if the computation is long and complex. If you have not given yourself enough practice here, you will struggle through the rest of this course. And the ones that come after. You may even fail.
So, right here, right now, resolve to do all of the problems in this chapter. Do them even if your instructor has not assigned them all. The more practice you get with this now, the less you will struggle later. Find other problems from other books and do them too. Make up your own problems. Practice until you can compute a differential with a quick glance.
This is important. We, the authors of this text, know what we are talking about. Listen to us. Your goal in this chapter is to completely internalize these differentiation rules. They need to become second nature to you. The work you do now, here, will pay off in the long run.
