Ch. 3.3, Rules of Differentiation

The Derivative of a function: If 𝑦 = 𝑓(𝑥) is a function, then 𝑓 ′ (𝑥) 𝑂𝑅 𝑑𝑦 𝑑𝑥 , the rate of change of 𝑦 with respect to 𝑥, is defined by 𝑓 ′ (𝑥) = lim ℎ→0 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ , if it exists. = the slope of the tangent line at (𝑥, 𝑓(𝑥)). = the rate of change of 𝑓(𝑥) with respect to 𝑥. Notations for the derivative: 𝑓 ′ (𝑥), 𝑑𝑦 𝑑𝑥 , 𝑑 𝑑𝑥 𝑓(𝑥), 𝐷𝑥 𝑓(𝑥) Differentiability and Continuity: Result: If a function is differentiable at 𝑥 = 𝑎, then it is continuous at 𝑥 = 𝑎. The derivative 𝑓 ′ exists when the function 𝑓 satisfies each of the following conditions at a point. 1. 𝑓 is continuous. 2. 𝑇he graph of 𝑓 is smooth, and 3. The graph of 𝑓 does not have a vertical tangent line. The derivative 𝑓 ′ does not exist when at least one of the following conditions is satisfied for 𝑓(𝑥) at a point 𝑥 = 𝑎. 1. 2. 3. 𝑓 is discontinuous at 𝑥 = 𝑎. The graph of 𝑓 has a sharp corner at 𝑥 = 𝑎 , or 𝑓 has a vertical tangent line at 𝑥 = 𝑎. 3.3: Rules of Differentiation Rule 1: Derivative of a constant: 𝑑 𝑑𝑥 (𝑐) = 0 Examples: a) 𝑑 𝑑𝑥 (7) = 0 b) 𝑑 𝑑𝑥 (𝜋) = 0 Rule 2: The Power property: 𝑑 𝑑𝑥 𝑑 (𝑥 𝑛 ) = 𝑛𝑥 𝑛−1 , 𝑑𝑥 (𝑒 𝑥 ) = 𝑒 𝑥 , 𝑑 𝑑𝑥 (𝑒 𝑛𝑥 ) = 𝑛𝑒 𝑛𝑥 Examples: a) 𝑓(𝑥) = 𝑥 3 3 e) 𝑓(𝑥) = √𝑥 1 b) 𝑓(𝑥) = 𝑥 c) 𝑓(𝑥) = f) 1 𝑦= 1 3 √𝑥 g) 𝑦 = 𝑥 𝑥2 5⁄ 2 1 d) 𝑦 = √𝑥 h) 𝑦 = 𝑥 −5 Rule 3: Constant Multiple Rule: 𝑑 𝑑𝑥 [𝑐𝑓(𝑥)] = 𝑐 𝑑 𝑑𝑥 [𝑓(𝑥)] Examples: a) If 𝑓(𝑥) = 5𝑥 3 , then 𝑓 ′ (𝑥) = 15𝑥 2 b) If 𝑓(𝑥) = 3 √𝑥 then 𝑓 ′ (𝑥) = − 3 3 2𝑥 2 Rule 4: The Sum/ difference Rule: 𝑑 𝑑 𝑑 [𝑓(𝑥) ± 𝑔(𝑥)] = [𝑓(𝑥)] ± [𝑔(𝑥)] 𝑑𝑥 𝑑𝑥 𝑑𝑥 Example: 1. Find the derivatives of the following functions: a) 𝑓(𝑥) = 2𝑥 4 − 𝑥 3 + 8 𝑡2 5 b) 𝑔(𝑡) = c) 𝑔(𝑥) = √𝑥 + 5 + 𝑡3 1 √𝑥 1 d) 𝑘(𝑥) = 𝑥 2 + 𝑥 2 e. Find the slopes of the tangent lines to the graph of 𝑓(𝑥) = 𝑥 2 + 1 at the points (0, 1) and (−1, 2). 2. Find the derivative of the function with respect to 𝑥: 𝒇(𝑥) = 𝑥 3 − 12𝑥 a) Find the points on the graph where the tangent line is horizontal. b) Write an equation of the tangent line at these points. **If a function is differentiable at a point c, then it is continuous at that point. **It is possible for a function to be continuous at c and not be differentiable at c. 3.3 Rules of Differentiation 𝒅 Rule 1: Derivative of a constant: 𝒅 Rule 2: The Power rule: 𝒅𝒙 𝒅𝒙 (𝒄) = 𝟎 (𝒙𝒏 ) = 𝒏𝒙𝒏−𝟏 Rule 3: Derivative of a constant times a Function: 𝒅 𝒅𝒙 [𝒄𝒇(𝒙)] = 𝒄 Rule 4: The Sum Rule: 𝒅 𝒅𝒙 𝒅 𝒅𝒙 [𝒇(𝒙)] [𝒇(𝒙) ± 𝒈(𝒙)] = 𝒅 𝒅𝒙 [𝒇(𝒙)] ± 𝒅 𝒅𝒙 [𝒈(𝒙)] Rule 5: The Product Rule: 𝒅 [𝒇(𝒙)𝒈(𝒙)] = 𝒇(𝒙)𝒈′ (𝒙) + 𝒈(𝒙)𝒇′ (𝒙) 𝒅𝒙 Examples: Use the product rule to differentiate each of the following functions. 𝑓(𝑥) = (2𝑥 2 + 1)(𝑥 2 − 2) a) 1 (𝑥 2 + 𝑒 𝑥 ) 𝑏) 𝑦 = 𝑒 2𝑥 b) 𝑎) 𝑦 = c) 𝑦 = (𝑥 2 + 1) (𝑥 + 5 + 𝑥) 𝑥 1 Rule 6: The Quotient Rule: 𝑑 𝑓(𝑥) 𝑔(𝑥)𝑓 ′ (𝑥) − 𝑓(𝑥)𝑔′ (𝑥) [ ]= [𝑔(𝑥)]2 𝑑𝑥 𝑔(𝑥) Examples: Find the derivative of the function. 2𝑥+5 1. 𝑓(𝑥) = 3𝑥−2 𝟓𝒙𝟐 −𝟏 2. 𝒉(𝒙) = 𝟐𝒙𝟑 +𝟑 3. 𝒇(𝒙) = 𝒙𝟐 −𝟑𝒙 𝒙−𝟏 Rule 7: The Extended Power Property: 𝑑 𝑑 [𝑓(𝑥)]𝑛 = 𝑛[𝑓(𝑥)]𝑛−1 . [𝑓(𝑥)] 𝑑𝑥 𝑑𝑥 Examples: a) 𝑓(𝑥) = (𝑥 3 + 5)2 c) 𝑓(𝑥) = (2𝑥 − 1)4 + (2𝑥 + 1)4 d) 𝑓(𝑥) = b) 𝑓(𝑥) = √1 − 𝑥 2 1 𝑥 3 +4𝑥 2𝑥+1 e) 𝑦 = √2𝑥−1