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2-2: Basic Differentiation Rules Objectives: 1. To derive and use the basic rules of differentiation Assignment: • P. 115-118: 1, 3-23 odd, 27, 29, 33, 39-61 eoo, 63, 69, 70, 83-89, 92, 111, 113, 115, 116 Warm-Up Use the limit definition to find the derivative of 𝑦 = 𝑐, where 𝑐 ∈ ℝ. Constant Rule Power Rule Constant Multiple Rule Objective 1 You will be able to derive and use the basic rules of differentiation Sum/Difference Rule Sine/Cosine Rules Local Linearity 𝑦 = 𝑥2 𝑦 = 𝑥3 𝑦 = 𝑥4 Derivative Function 𝑦=𝑥 ⋯ 𝑦′ = 1 𝑦′ = 2𝑥 𝑦′ = 3𝑥 2 𝑦′ = 4𝑥 3 ⋯ Pascal’s Triangle 𝑥+𝑐 𝑥+𝑐 4 2 𝑥 2 + 2𝑥𝑐 + 𝑐 2 𝑥 4 + 4𝑥 3 𝑐 + 6𝑥 2 𝑐 2 + 4𝑥𝑐 3 + 𝑐 4 Exercise 1 Use Pascal’s Triangle to expand 𝑥 + 𝑐 3 . Binomial Theorem In the expansion of 𝑥 + 𝑐 𝑥+𝑐 𝑛 𝑛 = 𝑥 + 𝑛𝑥 𝑛−1 𝑛 𝑛! 𝑐 + ⋯+ 𝑥 𝑛−𝑟 𝑐 𝑟 + ⋯ + 𝑛𝑥𝑐 𝑛−1 + 𝑐 𝑛 𝑛 − 𝑟 ! 𝑟! This is the number of combinations of 𝑛 objects taken 𝑟 at a time. Pascal’s Triangle For 𝑟 = 4: 10! = 10 − 4 ! 4! 10! = 6! 4! 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6! = 6! 4 ∙ 3 ∙ 2 ∙ 1 210 Exercise 2 Show that 𝑑 𝑑𝑥 𝑥 𝑛 = 𝑛𝑥 𝑛−1 . Constant and Power Rules Constant Rule Power Rule 𝑑 𝑐 =0 𝑑𝑥 𝑑 𝑛 𝑥 = 𝑛𝑥 𝑛−1 𝑑𝑥 𝑐, 𝑛 ∈ ℝ Exercise 3a Find the derivative of the following. 1. 𝑦 = 7 2. 𝑠 𝑡 = −3 3. 𝑑 𝑑𝑥 𝑘𝜋 2 Exercise 3b Find the derivative of the following. 1. 𝑓 𝑥 = 𝑥 3 2. 𝑔 𝑥 = 3 𝑥 3. 𝑦 = 1 𝑥2 Exercise 4 Find an equation of the tangent line to the graph of 𝑓(𝑥) = 𝑥 2 when 𝑥 = −2. Exercise 5 Find 𝑑 𝑑𝑥 𝑐𝑓(𝑥) . Constant Multiple Rule Constant Multiple Rule 𝑑 𝑐𝑓(𝑥) = 𝑐𝑓′(𝑥) 𝑑𝑥 𝑐∈ℝ Exercise 6 Find the derivative of the following. 1. 𝑦 = 2 𝑥 2. 𝑓 𝑡 = 4𝑡 2 5 3. 𝑦 = 1 3 2 𝑥2 Exercise 7 Find 𝑑 𝑑𝑥 𝑓 𝑥 +𝑔 𝑥 . Sum and Difference Rule The sum or difference of two differentiable functions 𝑓 and 𝑔 is itself differentiable. Moreover, the derivative of 𝑓 + 𝑔 or 𝑓 − 𝑔 is the sum or difference of the derivatives of 𝑓 and 𝑔. Sum Rule Difference Rule 𝑑 𝑓 𝑥 + 𝑔(𝑥) = 𝑓 ′ 𝑥 + 𝑔′(𝑥) 𝑑𝑥 𝑑 𝑓 𝑥 − 𝑔(𝑥) = 𝑓 ′ 𝑥 − 𝑔′(𝑥) 𝑑𝑥 Exercise 8 Find the derivative of the following. 1. 𝑓 𝑥 = 𝑥 3 − 4𝑥 + 5 2. 𝑔 𝑥 = 𝑥4 − 2 + 3𝑥 3 − 2𝑥 Exercise 9 Use local linearity to approximate the derivative of 𝑓(𝑥) = sin 𝑥. Sine and Cosine Rules 𝑑 sin 𝑥 = cos 𝑥 𝑑𝑥 𝑑 cos 𝑥 = − sin 𝑥 𝑑𝑥 Exercise 10 Show that 𝑑 𝑑𝑥 sin 𝑥 = cos 𝑥. Exercise 11 Find the derivative of the following. 1. 𝑦 = sin 𝑥 2 2. 𝑦 = 𝑥 + cos 𝑥 2-2: Basic Differentiation Rules Objectives: 1. To derive and use the basic rules of differentiation Assignment: • P. 115-118: 1, 3-23 odd, 27, 29, 33, 39-61 eoo, 63, 69, 70, 83-89, 92, 111, 113, 115, 116