More About Derivatives (Part 2)
Notation: The derivative of a function is also denoted as
This notation comes from the fact that when you compute the derivative, you compute
The term is usually referred as , and the term is referred as . So, sometimes, in some books (specially Physics books) you're going to find the definition
Theorems to calculate DerivativesNow it is the time to introduce the heavy artillery. In practice, you won't be computing the limit
very often. It is very important to know how to do it that way, but most of the times it won't be necessary.
Examples of Derivatives
Example: Compute the derivative of the function .
Solution: What do we do here, do we apply the limit to compute the derivative?? Well, your line of reasoning should be the following: The function corresponds to the sum of and . The intuition is that if I could compute the derivative of each term separately, then I could simplify the calculation.
In other words, if I knew what is the derivative of , and if I also knew what is the derivative of , then I should know what is the derivative of .....
In fact, you do. We have the following theorem:
Theorem: The Derivative of the Sum of Two Functions
Assume that and are differentiable at (that means that the derivative exists at that point). Then, we have that
In other words, the derivative of the sum is the sum of the derivatives (These are not empty words, they really describe the result accurately). This is usually referred as the Linearity Property of the derivative
Now we show a result which is one of the most important derivative rules, that will help us to compute a lot of derivatives:
Theorem: The following holds true for all :
Proof: We won't do anything too deep, just to save you from fatal boredom, but let's just do this one to get a feel of it. By definition
So, let's come back to the problem of finding the derivative of . Using the Linearity of derivatives we find that
Let's recall that , so applying that to the case and respectively we get the previous result. The Linearity property can be written in a more general way:
Theorem: Assume that and are differentiable at and and are constants. Then
Below we show an example of how to apply this result:
Example: Compute the derivative of the function .
Solution: Using Linearity, we get that