We want to use theorems, referred to as " chain rules ", to get to the idea of gradient vector.

There are , as would be expected, many other uses. Recall that " chain rule " is a designation for theorems about derivatives of functions which are compositions. There are an infinity of such theorems because there are an infinity of possible counts on the number of variables involved in the functions under consideration. Consequently, you will "see" expressions like the following as we do the introduction to this idea :

These formulas may not make sense to us at first sight, but they represent types of forms we encounter when doing " chain rule " theorems .

Let's look at how they might be derived.

See13.5 1st item Chain Rules

See 13.5 Example 1 Using the Chain Rule ( one independent variable )

See 13.5 Example 2 An application to related rates

Our next example has us computing partial derivatives by simply substituting to remove the composition, Then, for contrast, I've used a theorem to do the same problem. Notice the  results are the same.

See 13.5 Example 3

Now by theorem :

=2y(2s) + 2x(1/t) then

Look at the following solved problems in the Exercises at the end of 13.5 : 1-15 odd and 41. Then
do the following unsolved problems as homework: 2-14 even and 42 .