The Chain Rule

The Chain Rule

The Chain Rule is what allows us to extend the simple rules for differentiation to more complicated functions. The Chain Rule states that

Another way to look at it is to say that suppose is a function of and is a function of . Then

Leibniz' notation makes it look like the quantities are "cancelling out" but that's not what is happening. That's the beauty of the notation, you can always pretend that's what is going on and you'll get the right answer!

Look at what this implies in practical terms of calculating derivatives.

Example: To find

we can see this as

where

. The Chain Rule says this equals

Green Question: Evaluate each of the following derivatives.

(A) (B) (C) (D)

STEP 2000, Math I, #7: Let

is a constant. Show that, if , then for all .

[Hint: The problem is to show that does not exceed a certain value. You could consider what the maximum value of might be.]

Skill Set

Chain Rule

No. 1