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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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 Thomson Learning, Inc.

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 second function Derivative of the first 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 division of Thomson Learning, Inc.

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

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

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

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Higher Derivatives The second derivative of a function f is the derivative of 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 Learning, Inc.

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