1. the Chain Rule
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AP CALCULUS LECTURE NOTES MR. RECORD Section Number: Topics: The Chain Rule Day: 15 2.4 A
1. The Chain Rule
Perhaps the best way to introduce this new differentiating technique is to show you the types of functions that would require its use. The table below illustrates pairs of similar functions that can be differentiated with and without the Chain Rule. Without the Chain Rule With the Chain Rule
Example 1: Decompositions of a Composite Function. Complete the table. a.
b.
c.
d.
Example 2: Find for Example 3: Find for each of the following. a.
b.
c.
AP CALCULUS LECTURE NOTES MR. RECORD Section Number: Topics: The Chain Rule Day: 16 - Differentiating tan(x), cot(x) sec(x) and csc(x) 2.4 B - Nested Chain Rules 2. Derivatives of The Other Trigonometric Functions
Recall from Section 2.2,
Now, we will take a look at the derivatives of the other four trigonometric functions. Example 4: Prove
Example 5: Find the derivative of each of the following. a. b.
Example 6: Differentiate both forms of
3. Trigonometric Functions and the Chain Rule
The “Chain Rule versions” of the derivatives of the six trigonometric functions are as follows
Example 7: Find the derivative of each trigonometric function. a. b. c.
4. Trigonometric Functions That Require Repeated Use of the Chain Rule Example 8: Find the derivative of each trigonometric function. a. b.
c.
5. Nested Product/Quotient and Chain Rules
Lastly, we will see how the Chain Rule can be used in conjunction with the Product and Quotient Rules. It takes a bit more planning and organization to take the derivative correctly, but it’s really not too bad. Furthermore, there is a special technique used to simplify these answers.
Example 9: Differentiate and simplify .
Example 10: Differentiate and simplify . Example 11: Differentiate and simplify .