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Understanding Chain & General Power Rules for Calculus

Learn how the Chain Rule and General Power Rule work in calculus with examples and explanations. Discover how to calculate derivatives efficiently using these rules to simplify complex functions.

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Understanding Chain & General Power Rules for Calculus

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  1. The Chain & General Power RuleSection 2.4By Liz Labuzienski, Soee Kwon and Sue Kim

  2. Chain Rule • If y changes dy/du times as fast as u, and u changes du/dx times as fast as x, then y changes dy/du · du/dx times as fast as x. • In other words, dy/dx = dy/du · du/dx • The Chain Rule:

  3. Examples

  4. Examples

  5. General Power Rule • If y = (u(x))n, where u(x) is a differentiablefunctionand nis a rational number,then or, equivalently,

  6. Examples • Note: If u = 3x – 2x2, then f(x) = u3. This means that the derivative can be rewritten as 3u2u’, which may help in the simplifying process.

  7. Examples

  8. Trigonometric Functions and the Chain Rule

  9. Example • Note: In this example, the function is a composition of three functions, so the Chain Rule will need to be applied twice.

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