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Techniques of Differentiation

Techniques of Differentiation. I. Positive Integer Powers, Multiples, Sums, and Differences. A.) Th : If f ( x ) is a constant, B.) Th : The Power Rule: If n is a positive integer and,

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Techniques of Differentiation

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  1. Techniques of Differentiation

  2. I. Positive Integer Powers, Multiples, Sums, and Differences A.) Th: If f(x) is a constant, B.) Th: The Power Rule: If n is a positive integer and, C.) Th: The Constant Multiple Rule: If k is a real number and f is differentiable at x, then

  3. D.) Th: The Sum/Difference Rule: If f and g are differentiable functions of x at x, then their sum and their difference are differentiable at ever point where f and g are differentiable

  4. II. Examples A.) Find the following derivatives:

  5. III. Derivative Notation for Functions of x A.) Given u as a function of x, the derivative of u is written as follows:

  6. IV. Product Rule A.) Th:

  7. B.) Use the product rule to find the derivatives of the following functions:

  8. C.) Let y = uvbe the product of the functions u and v. Find y’(2) if u(2) = 3, u’(2) = -4, v(2) = 1, and v’(2) = 2.

  9. V. Quotient Rule A.) Th:

  10. B.) Use the quotient rule to find the derivatives of the following functions:

  11. VI. Negative Exponent Thm: A.) For any integer n ≠ 0, B.) Find the following derivative using both the quotient rule and the negative exponent theorem.

  12. VI. Tangent Lines Ex: Find the instantaneous rate of change of the following functions at the given x values. Then, give the equation of the tangent line at that point. • Find ordered pair: • Find derivative: • Substitute x=3 into derivative: This is the slope. • Find the equation:

  13. Derivatives of Trigonometric Functions

  14. I. Sin, Cos, Tan A.) B.) C.)

  15. D.)

  16. II. Examples A.) Find the following derivatives:

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