Differentiation 3 Basic Rules of Differentiation The Product

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Differentiation 3 • Basic Rules of Differentiation • The Product and Quotient Rules •

Differentiation 3 • Basic Rules of Differentiation • The Product and Quotient Rules • The Chain Rule Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Basic Differentiation Rules 1. Ex. 2. Ex. Copyright © 2006 Brooks/Cole, a division of

Basic Differentiation Rules 1. Ex. 2. Ex. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Basic Differentiation Rules 3. Ex. 4. Ex. Copyright © 2006 Brooks/Cole, a division of

Basic Differentiation Rules 3. Ex. 4. Ex. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

More Differentiation Rules 5. Product Rule Ex. Derivative of the first function Derivative of

More Differentiation Rules 5. Product Rule Ex. Derivative of the first function Derivative of the second function Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

More Differentiation Rules 6. Quotient Rule Sometimes remembered as: Copyright © 2006 Brooks/Cole, a

More Differentiation Rules 6. Quotient Rule Sometimes remembered as: Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

More Differentiation Rules 6. Quotient Rule (cont. ) Ex. Derivative of the numerator Derivative

More Differentiation Rules 6. Quotient Rule (cont. ) Ex. Derivative of the numerator Derivative of the denominator Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

More Differentiation Rules 7. The Chain Rule Note: h(x) is a composite function. Another

More Differentiation Rules 7. The Chain Rule Note: h(x) is a composite function. Another Version: Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

More Differentiation Rules The Chain Rule leads to The General Power Rule: Ex. Copyright

More Differentiation Rules The Chain Rule leads to The General Power Rule: Ex. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Chain Rule Example Ex. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Chain Rule Example Ex. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Chain Rule Example Ex. Sub in for u Copyright © 2006 Brooks/Cole, a division

Chain Rule Example Ex. Sub in for u Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Higher Derivatives The second derivative of a function f is the derivative of f

Higher Derivatives The second derivative of a function f is the derivative of f at a point x in the domain of the first derivative. Derivative Notations Second Third Fourth nth Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example of Higher Derivatives Given find Copyright © 2006 Brooks/Cole, a division of Thomson

Example of Higher Derivatives Given find Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example of Higher Derivatives Given find Copyright © 2006 Brooks/Cole, a division of Thomson

Example of Higher Derivatives Given find Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.