Product & Quotient Rule - PowerPoint PPT Presentation

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Product & Quotient Rule

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  1. Product & Quotient Rule Higher Order Derivatives

  2. The Product Rule • Theorem. Let f and g be differentiable functions. Then the derivative of the product fg is • (fg) '(x) = f(x) g '(x) + g(x) f '(x) In other words, first times the derivative of the second plus second times the derivative of the first.

  3. Using the Product Rule • Example:

  4. Product Rule • Example:

  5. Product Rule • Example

  6. Quotient Rule • Theorem. Let f and g be differentiable functions. Then the derivative of the quotient f/g is In other words, low d high minus high d low over low low.

  7. Quotient Rule • Example • Find the derivative of

  8. Rewriting Before Differentiating • Example:

  9. Derivatives of Trigonometric Functions

  10. Proof of Derivative of Tangent • Considering

  11. Differentiating Trig Functions • y = x – tan x • y = x sec x y’ = x(sec x tan x) + sec x (1) = sec x (x tan x + 1)

  12. Differentiating Trigonometric Functions

  13. Higher Order Derivatives Applications: Finding Acceleration Due to Gravity Population Growth p. 125 problem 79 Any time we are asked to find the rate at which a rate is changing, this is a second derivative.