2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 2.3 Product and Quotient Rules, Trig, and Higher Order Derivatives Objective: You will be able to: • use the Product Rule • use the Quotient Rule • find the derivative of all trigonometric functions • find a higher order derivative of a function 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 Stand and Deliver Product Rule d [f(x)g(x)] =[f(x)g '(x) + g(x)f '(x)] dx "first times derivative of the second plus second times the derivative of the first" 2.3 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 Examples 1. f(x) = (3x)(x2 + 1) one option (distribute and use power rule) another option (product rule) 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 Examples 2. f(x) = x cosx no choice, must use product rule 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 Your turn! y = (x2 + 2x)(x + 1) 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 Stand and Deliver Calculus the Musical: Quotient Rule Quotient Rule [ [ = d f(x) dx g(x) g(x)f '(x) f(x)g'(x) [g(x)]2 "lo D hi, minus hi D lo, over lo lo" 2.3 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 Examples 1. g(x) = 3x 1 x2 + 3 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 2. y = x2 +3x 6 trick question don't need quotient rule 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 3. Use the Quotient Rule to find d [tan x] dx 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 Stand and Deliver Calculus the Musical: Triggy Rules now add other 4 trig functions 2.2/2.3 Derivative of Trig. Functions d [tan x] = sec2 x d [sin x] = cos x dx dx d [cot x] = csc2 x d [cos x] = sin x dx dx d [sec x] = sec x tan x dx d [csc x] = csc x cot x dx 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 Notice Patterns: the derivatives of the "c's" always negative (cosine, cotangent, and cosecantall have negative derivatives) 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 Warm Ups find the derivative... 1. y = x tan x 2. f(x) = cos x sin x 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 Stand and Deliver Higher Order Derivatives 2.3 s(t) = position v(t) = s'(t) = velocity a(t) = v'(t) = s''(t) = acceleration speed = Iv(t)I 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 1. If the nth derivative of y is denoted as yn and y = -sin x, then y7 is the same as: (A) y (B) dy dx (C) d2y (D) d3y (E) none of the above dx2 dx3 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013 2. Use the table below to find the value of (A) 5/2 (B) -3/2 (C) -13 x 1 3 (D) 12 f(x) g(x) 4 7 2 4 at x = 3. (E) 21/2 f ' (x) g ' (x) 5 1 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook September 12, 2013