MIT OpenCourseWare ocw.mit 18.01 Single Variable Calculus Fall 2006

1. MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citingthese materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 4 Sept. 14, 2006 18.01 Fall 2006

2. Lecture 4ChainRule, and Higher Derivatives Lecture 4 Sept. 14, 2006 18.01 Fall 2006

3. Chain Rule We’ve got general procedures for differentiating expressions with addition, subtraction, and multiplication. What about composition? Example 1. Lecture 4 Sept. 14, 2006 18.01 Fall 2006

4. Chain Rule So, . To find , write Lecture 4 Sept. 14, 2006 18.01 Fall 2006

5. Chain Rule As too, because of continuity. So we get: In the example, So, Lecture 4 Sept. 14, 2006 18.01 Fall 2006

6. Chain Rule Another notation for the chain rule (or ) Example 1. (continued) Composition of functions . Note: Not Commutative! Lecture 4 Sept. 14, 2006 18.01 Fall 2006

7. Chain Rule Figure 1: Composition of functions:(f ⃘g)(x)=(g(x)) Lecture 4 Sept. 14, 2006 18.01 Fall 2006

8. Chain Rule Example 2. Let Lecture 4 Sept. 14, 2006 18.01 Fall 2006

9. Chain Rule Example 3. There are two ways to proceed. Lecture 4 Sept. 14, 2006 18.01 Fall 2006

10. Higher Derivatives Higher derivatives are derivatives of derivatives. For instance, if , then is the second derivative of f. We write Notations Higher derivatives are pretty straightforward ⇁ just keep taking the derivative! Lecture 4 Sept. 14, 2006 18.01 Fall 2006

11. Higher Derivatives Example. Start small and look for a pattern. The notation n! is called “n factorial” and defined by Lecture 4 Sept. 14, 2006 18.01 Fall 2006

12. Higher Derivatives Proof by Induction: We’ve already checked the base case (n = 1). Induction step: Suppose we know Show it holds for the case. Proved! Lecture 4 Sept. 14, 2006 18.01 Fall 2006