Rules for Derivatives

What you should see in the following table is that for each way that we have for putting functions together we have a corresponding rule for putting the derivatives of the functions together. Since every function is put together from the elementary functions, we see that once we know the derivatives of the elementary functions (and we do) and these rules we can compute the derivative of any function.

Note carefully that the derivatives do not necessarily go together the way that the functions do. They do in the first two rules but not in the others. However, the rules are the same no matter what the component functions. They are extremely systematic and are applied the same way now and forever. Once you learn how to use them you will always be able to compute the derivative of any function because every function is made up from the elementary functions.

All of these rules come from the definition of the derivative. We are not interested in their derivation, only how to use them. You will not have to derive them for the test.

You must commit this table to memory.

Let's see what is being said in words.

You should first compare the two tables. Make sure that you understand that they say the same things. The best way to learn the rules is to write down the corresponding functions in a particular problem as you say the words. If you do this consistently in your homework, then this part of the test should be relatively easy.

You should strive to work the problems in the same way that I do. Don't look for short cuts. Make your work look like mine. I work the problems in a way that I believe will help you to maximize your grade in the course. In fact, later on I will state that certain things must be done in a particular way. Don't do things in your head, get it down on paper. You must practice. Once its down on the test, that's it. No reprieve.

For the remainder of this section we will practice using the first two rules; the constant times a function rule and the sum rule. In the next section, we do the product rule and the quotient rule. Then the composition (chain) rule. Finally we do the derivatives of piecewise functions.

Examples

Constant times a function rule.

Sum rule.

We can now combine the two rules to compute the derivatives of functions that are sums of constants times functions.

Notice carefully how I use just one rule at a time. When I'm using the sum rule, that's all that I do even if it means copying some terms without change. Then I use the constant times a function rule. Now the only thing left to do is to compute the derivatives of the elementary functions. You should adopt this one step at a time mentality. It may seem trivial now but when things get tougher it will pay off. Don't be in such a hurry to "get the answer". It's the technique that I'm teaching here and its the technique that you will be tested on. If you don't learn it in these simple cases, it will be much more difficult to learn it when things are harder.

Homework

Constant times a function.

Sum of functions.

Sums of constants times functions.

Answers

Constant times a function # 1 - 5 with work.

Sum of functions # 1 - 5 with work.

Sums of constants times functions # 1 - 5 with work.