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Section 2.3

Section 2.3. Day 1. Product Rule. The product of two differentiable functions f and g is itself differentiable. Example 1. Find the derivative of h ( x ) = (3 x – 2 x 2 )(5 + 4 x ) using the Product Rule. Let f ( x ) = 3 x – 2 x 2 and g ( x ) = 5 + 4 x. Example 2.

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Section 2.3

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  1. Section 2.3 Day 1

  2. Product Rule • The product of two differentiable functions f and g is itself differentiable.

  3. Example 1 • Find the derivative of • h(x) = (3x – 2x2)(5 + 4x) • using the Product Rule. • Let f(x) = 3x – 2x2 and g(x) = 5 + 4x

  4. Example 2 • Find the derivative of y = 2x3cos x – 2sin x.

  5. The Product Rule can be extended to cover products involving more than two factors. For example, if f, g, and h are differentiable functions of x, then

  6. Quotient Rule • The quotient f/g of two differentiable functions f and g is itself differentiable at all values of x for which g(x) ≠ 0.

  7. Example 3 • Find the derivative of

  8. Example 4 • Find the equation of the tangent line to the graph of

  9. Rewrite f(x) by simplifying the complex fraction, then use the quotient rule.

  10. The equation of the tangent line at (−1, 1) is • y = 1.

  11. Sometimes it is easier to rewrite a quotient as a product of a constant and a function of x. When you can do this then use the Constant Multiple Rule instead of the Quotient Rule.

  12. Example 6

  13. Section 2.3 2nd Day

  14. Derivatives of Trigonometric Functions • Find the following derivatives using the quotient rule. Rewrite each in terms of sine and cosine.

  15. Example 1 • Find the derivative of the following • a. y = x – tan x • b. y = x sec x

  16. a. y = x – tan x • b. y = x sec x

  17. Higher Order Derivatives • A second derivative, f ′′(x), is the derivative of the first derivative of a function. • A third derivative, f′′′(x), is the derivative of the second derivative of a function. • A fourth derivative, f(4)(x), is the derivative of the third derivative of a function. • These are examples of higher order derivatives.

  18. Example 3 • Find the second and third derivatives of • f(x) = 2x3 – 3x2 + 4x – 5

  19. Acceleration • The acceleration function is found by differentiating the velocity function. Since the velocity function is the derivative of the position function, the acceleration function is the second derivative of the position function.

  20. Example 4 • Because the moon has no atmosphere, a falling object on the moon encounters no air resistance. In 1971, astronaut David Scott demonstrated that a feather and a hammer fall at the same rate on the moon. The position function for each of these falling objects is given by s(t) = -0.81t2 + 2 where s(t) is the height in meters and t is the time in seconds. What is the ratio of Earth’s gravitational force to the moon’s?

  21. Find the acceleration on the moon. • The acceleration due to gravity on the moon is -1.62 meters per second per second. The acceleration due to gravity on Earth is -9.8 meters per second per second.

  22. AP Questions

  23. Let f and g be differentiable functions with the following properties: • (i) g(x) > 0, for all x • (ii) f(0) = 1 • If h(x) = f(x)g(x) and h΄(x) = f ΄(x)g(x), then f(x) = • (A) f΄(x) (B) g(x) (C) ex • (D) 0 (E) 1 E

  24. What is the instantaneous rate of change at x = 2 of the function f given by

  25. HW: pp.126-129 (4-80 mult. of 4, 90, 94-102 even, 104, 106, 112, 116, 118)

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