Differentiation of a function was defined as a limit of a difference quotient in our first Calculus course.

See 2.1 Derivative of a Function Pg.99

We will use the limit of the difference quotient for differentiation of a Vector-Valued Function. However, this will result in a vector rather than a real number (scalar); this should be obvious to you when you consider that r(t) is a vector in this new definition of differentiation.

See 12.2 Differentiation of Vector-Valued Functions ( read the definiton)

It will help if we look at the idea of limit of a Vector Function for just a moment. We previously said that it would involve the limit work done in the first Calculus course but that it would be done to the "components" of the Vector Function. A quick "look" at the solutions to a two such problems may help:

See Exercises for Section 12.1 #69 Pg.839

See Exercises for Section 12.1 #71 Pg.839

This idea of limit allows us to "see" why differentiation of a vector function can be done by differentiating the component fuctions.

See 12.2 ( following the Definition) component-by-component basis

This explains Thm. 12.1 :

See 12.2 Differentiation of Vector -Valued Function( read  Theorem 12.1)

Finally, we have  Theorem 11.2 whose parts resemble previous theorems from the first Calculus I. We will not be doing enough work in this section to require us to use this theorem but do "look" at it:

See 12.2 Differentiation of Vector-Valued Function (read theorem 12.2)

Now "see" these examples of differentiation of vector functions:

See 12.2 Example 1

I include this next item because your authors mention the rules for differentiation in the appendix:

See Appendix C A27

Integration of Vector functions may be of the indefinite or the definite type, just as in Calculus I . We will have use for the indefinite integral in our work but both are mentioned in the text that follows:

See 12.2 Integration of Vector-Valued Functions

The significance of the indefinite integral is simply that the derivative function might be known and you are required to find vector functions that correspond to that derivative function.

See 12.2 Example 5

See 12.2 Example 7

Look at the following solved problems in the Exercises at the end of
12.2 : 1,3,7,11,13,15,17,19a,19b,37,39,41,45 Then do the following unsolved
problems as homework: 12,14,16,18,20a,20b,38,40,42,42