Lecture 4 Sept. 14, 2006 18.01 Fall 2006 1

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Lecture 4 Sept. 14, 2006 18.01 Fall 2006 2

Chain Rule

Higher Derivatives

* Lecture 4ChainRule, and Higher Derivatives

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We’ve got general procedures for differentiating expressions with addition, subtraction, and multiplication. What about composition?

Example 1.

* Chain Rule

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So, . To find , write

* Chain Rule

 

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As too, because of continuity. So we get:

In the example,

So,

* Chain Rule

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Another notation for the chain rule

(or )

Example 1. (continued) Composition of functions .

Note: Not Commutative!

* Chain Rule

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* Chain Rule

Figure 1: Composition of functions: (f ⃘g)(x)=(g(x))

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Example 2.

Let

* Chain Rule

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Example 3.

There are two ways to proceed.

 

* Chain Rule

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Higher derivatives are derivatives of derivatives. For instance, if , then is the second

derivative of f. We write

Notations

 

* Higher Derivatives

Higher derivatives are pretty straightforward just keep taking the ⇁derivative!

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Example.

Start small and look for a pattern.

The notation n! is called “n factorial” and defined by

 

* Higher Derivatives

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Proof by Induction: We’ve already checked the base case (n = 1).

Induction step: Suppose we know Show it holds for the case.

Proved!

 

* Higher Derivatives