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 1.y = 5x³ + 4x² + 6x y =  x² + 1 Find second derivatives of the following: 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)

Example 2 3.Find the fourth derivatives of f(x) = (1/90)x 10 + (1/60)x 5 4.Find a formula for f n (x) where f(x) = x -2 f’(x) = (10/90)x 9 + (5/60)x 4 f’’(x) = (90/90)x 8 + (20/60)x 3 f’’’(x) = (720/90)x 7 + (60/60)x 2 F’’’’(x) = (5040/90)x 6 + (120/60)x = 56x 6 + 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 f n (x) = (-1) n (n+1)! x -(2-n)

Example 3 5.y = (3x³ - 4x² + 7x - 9) 6.f(x) = e 2x+7 Find the third derivatives of the following: f’(x) = 2e 2x+7 f’’(x) = 4e 2x+7 f’’’(x) = 8e 2x+7 y’(x) = 9x² - 8x + 7 y’’(x) = 18x - 8 y’’’(x) = 18

Chain Rule Revisited FunctionDerivativeFunctionDerivative y = x n y’ = n x n-1 y = u n y’ = n u n-1 u’ y = e x y’ = e x y = e u y’ = u’ e u y = sin xy’ = cos xy = sin uy’ = u’ cos u y = cos xy’ = -sin xy = cos uy’ = -u’ sin u y = tan xy’ = sec² xy = tan uy’ = u’ sec² u y = cot xy’ = -csc² x y = cot uy’ = u’ csc² u where u is a function of x (other than just u=x) and u’ is its derivative

Derivatives FAQ 1)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). 2)Does the argument in a trig function ever get changed? No. The item (argument) inside the trig function never changes while taking the derivative. 3)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. 4)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.

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