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Lesson 3-7

Lesson 3-7. Higher Order Deriviatives. Objectives. Find second and higher order derivatives using all previously learned rules for differentiation. Vocabulary. Higher order Derivative – taking the derivative of a function a second or more times. Example 1.

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Lesson 3-7

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  1. Lesson 3-7 Higher Order Deriviatives

  2. Objectives • Find second and higher order derivatives using all previously learned rules for differentiation

  3. Vocabulary • Higher order Derivative – taking the derivative of a function a second or more times

  4. Example 1 Find second derivatives of the following: • y = 5x³ + 4x² + 6x + 3 • y = x² + 1 y’(x) = 15x² + 8x + 6 y’’(x) = 30x + 8 y’(x) = ½ (x² + 1)-½ (2x) = x / (x² + 1)½ (1)(x² + 1)½ - x ½ (x² + 1) -½ (2x) y’’(x) = -------------------------------------------- (x² + 1)

  5. Example 2 Find the fourth derivatives of f(x) = (1/90)x10 + (1/60)x5 Find a formula for f n(x) where f(x) = x-2 f’(x) = (10/90)x9 + (5/60)x4 f’’(x) = (90/90)x8 + (20/60)x3 f’’’(x) = (720/90)x7 + (60/60)x2 F’’’’(x) = (5040/90)x6 + (120/60)x = 56x6 + 2x f’(x) = (-2)x-3 f’’(x) = (-2)(-3)x-4 f’’’(x) = (-2)(-3)(-4)x-5 f’’’’(x) = (-2)(-3)(-4)(-5)x-6 fn(x) = (-1)n(n+1)! x-(2-n)

  6. Example 3 Find the third derivatives of the following: y = (3x³ - 4x² + 7x - 9) f(x) = e2x+7 y’(x) = 9x² - 8x + 7 y’’(x) = 18x - 8 y’’’(x) = 18 f’(x) = 2e2x+7 f’’(x) = 4e2x+7 f’’’(x) = 8e2x+7

  7. Chain Rule Revisited FunctionDerivativeFunctionDerivative y = xn y’ = n xn-1 y = un y’ = n un-1 u’ y = ex y’ = ex y = eu y’ = u’ eu y = sin x y’ = cos x y = sin u y’ = u’ cos u y = cos x y’ = -sin x y = cos u y’ = -u’ sin u y = tan x y’ = sec² x y = tan u y’ = u’ sec² u y = cot x y’ = -csc² x y = cot u y’ = u’ csc² u where u is a function of x (other than just u=x) and u’ is its derivative

  8. Derivatives FAQ • When do I stop using the chain rule?Answer: when you get something that you can take the derivative of without having to invoke the chain rule an additional time (like a polynomial function). • Does the argument in a trig function ever get changed?No. The item (argument) inside the trig function never changes while taking the derivative. • Does the exponent always get reduced by one when we take the derivative?Only if the exponent is a constant! If it is a function of x, then it will remain unreduced and you have to use another rule instead of simple power rule. • When do I use the product and quotient rules?Anytime you have a function that has pieces that are functions of x in the forms of a product or quotient.

  9. Summary & Homework • Summary: • Derivative of Derivatives • Use all known rules to find higher order derivatives • Homework: • pg 240 - 242: 5, 9, 17, 18, 25, 29, 49, 57

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