University:
Glendale Community College
Course:
MAT142 | College Mathematics
Academic year:

2019
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60

Pages:

20
Author:

boltakvf

Section 2.2 Derivatives of Products and Quotients (Minimum Homework: all odds) We will learn two new rules to find derivatives in section 2.2. The Product Rule – which is used when finding a derivative of problem with multiplication of two factors, both of which contain a variable. The Quotient Rule – which is used when finding the derivative of a fraction that has a variable in the denominator. Here are the rules and a short description of the symbols: The Product Rule: If 𝑓 𝑎𝑛𝑑 𝑔 are both differentiable functions, then: 𝑑 𝑑 𝑑 (𝑓(𝑥 ) ∗ 𝑔(𝑥 )) = 𝑓(𝑥 ) ∗ 𝑔(𝑥 ) + 𝑔(𝑥 ) ∗ 𝑓(𝑥) 𝑑𝑥 𝑑𝑥 𝑑𝑥 Product Rule (derivative equals) (𝑓𝑖𝑟𝑠𝑡 𝑓𝑎𝑐𝑡𝑜𝑟)(𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑠𝑒𝑐𝑜𝑛𝑑 𝑓𝑎𝑐𝑡𝑜𝑟) + 𝑠𝑒𝑐𝑜𝑛𝑑 𝑓𝑎𝑐𝑡𝑜𝑟(𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑓𝑎𝑐𝑡𝑜𝑟) The Quotient Rule If 𝑓 𝑎𝑛𝑑 𝑔 are both differentiable functions, then: 𝑑 𝑑 𝑔(𝑥 ) ∗ 𝑓(𝑥 ) − 𝑓(𝑥 ) ∗ 𝑔(𝑥) 𝑑 𝑓(𝑥 ) 𝑑𝑥 𝑑𝑥 ( )= (𝑔(𝑥))2 𝑑𝑥 𝑔(𝑥) Quotient Rule (derivative equals) 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟(𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟) − 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟(𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟) (𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟)2 Example : Find the derivative using the Product rule. 𝑓(𝑥 ) = (5𝑥 + 6)(𝑥 2 − 3𝑥) First: Determine the two factors. (parentheses are not required) First factor: 5𝑥 + 6 Second Factor: 𝑥 2 − 3𝑥 Place them in the top row of a table: First factor 5𝑥 + 6 Second Factor 𝑥 2 − 3𝑥 Second: find the derivative of each factor and put the derivative in the second row. First factor 5𝑥 + 6 Derivative 5 Second Factor 𝑥 2 − 3𝑥 Derivative 2𝑥 − 3 Third: Cross multiply top down and bottom up. First factor 5𝑥 + 6 Derivative 5 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 Second Factor 𝑥 2 − 3𝑥 Derivative 2𝑥 − 3 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 (5𝑥 + 6)(2𝑥 − 3) 5(𝑥 2 − 3𝑥) Fourth: Add the expressions in the bottom row to find the derivative. 𝑓 ′ (𝑥 ) = (5𝑥 + 6)(2𝑥 − 3) + 5(𝑥 2 − 3𝑥) Fifth: Simplify 𝑓 ′ (𝑥 ) = 10𝑥 2 − 15𝑥 + 12𝑥 − 18 + 5𝑥 2 − 15𝑥 Answer: 𝑓 ′ (𝑥 ) = 15𝑥 2 − 18𝑥 − 18 𝑜𝑟 3(5𝑥 2 − 6𝑥 − 6) Example: Find the derivative using the Quotient rule. 𝑓(𝑥 ) = 3𝑥 2𝑥+5 First: Create a table. Put the denominator in the top left position, the numerator in the top right position. Denominator 2𝑥 + 5 Numerator 3𝑥 Second: Find the derivative of each and put the result in the second row. Denominator 2𝑥 + 5 Numerator 3𝑥 Derivative 2 Derivative 3 Third: Cross multiply top down and bottom up. Denominator 2𝑥 + 5 Numerator 3𝑥 Derivative 2 Derivative 3 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 (2𝑥 + 5)3 = 6𝑥 + 15 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 2(3𝑥 ) = 6𝑥 Fourth: create a fraction. Place the expressions in the numerator with a subtraction between. Place the square of the denominator in the denominator. 𝑓 ′ (𝑥 ) = 6𝑥+15−6𝑥 (2𝑥+5)2 Fifth: Simplify 𝑓 ′ (𝑥 ) = 15 (2𝑥+5)2 15 Answer: 𝑓 ′ (𝑥 ) = (2𝑥+5)2 Example: 𝑓(𝑥 ) = (5𝑥 + 6)(𝑥 2 − 3𝑥) ; x = 2 a) Find the slope of the tangent line to the graph of the function for the given value of x. b) Find the equation of the tangent line to the graph of the function for the given value of x. a) Slopes of tangent lines can be found by substituting 𝑥 = 2 into the derivative. 𝑓 ′ (𝑥 ) = 15𝑥 2 − 18𝑥 − 18 (𝑓𝑟𝑜𝑚 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑒𝑥𝑎𝑚𝑝𝑙𝑒) 𝑚 = 𝑓 ′ (2) = 15(2)2 − 18(2) − 18 = 6 Answer 𝑚 = 6 b) Need to find y-coordinate of the point. 𝑦 = 𝑓(2) − (5(2) + 6)((2)2 − 3(2)) = −32 𝑝𝑜𝑖𝑛𝑡 (2, −32) 𝑠𝑙𝑜𝑝𝑒 𝑚 = 6 Equation of line 𝑦 − (−32) = 6(𝑥 − 2) 𝑦 + 32 = 6𝑥 − 12 Answer: 𝑦 = 6𝑥 − 44 #1-12: Use the product rule to find the derivatives of the following. 1) 𝑦 = (2𝑥 + 3)(3𝑥 – 4) 2) 𝑦 = (3𝑥 – 4)(5𝑥 – 8) First factor Second Factor Derivative Derivative 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 Answer: 𝑦 ′ = 30𝑥 − 44 3) 𝑓(𝑥) = (𝑥 – 2)(3𝑥 – 4) 4) 𝑦 = (𝑥 − 5)(3𝑥 2 + 7) First factor Second Factor Derivative Derivative 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 𝑎𝑛𝑠𝑤𝑒𝑟: 𝑑𝑦 𝑑𝑥 = 9𝑥 2 − 30𝑥 + 7 5) 𝑓(𝑥) = (𝑥 2 + 3𝑥 + 2)(3𝑥 – 5) 6) 𝑓(𝑥) = (3𝑥 2 + 6𝑥 – 2)(4𝑥 + 1) First factor Second Factor Derivative Derivative 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 𝑎𝑛𝑠𝑤𝑒𝑟: 𝑓 ′ (𝑥) = 36𝑥 2 + 54𝑥 − 2 7) 𝑔(𝑡) = (2𝑡 – 1)(3𝑡 + 5) 8) 𝑔(𝑡) = (3𝑡 2 + 5𝑡)(2𝑡 + 1) First factor Second Factor Derivative Derivative 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 𝑎𝑛𝑠𝑤𝑒𝑟: 𝑔′ (𝑡) = 18𝑡 2 + 26𝑡 + 5 9) 𝑦 = 3𝑥 2 (2𝑥 2 + 6𝑥 – 4) 10) 𝑦 = 4𝑥 3 (3𝑥 2 + 7𝑥 – 5) First factor Second Factor Derivative Derivative 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 𝑎𝑛𝑠𝑤𝑒𝑟: 𝑦 ′ = 4𝑥 2 (15𝑥 2 + 28𝑥 − 15) 11) 𝑦 = (3𝑥 4 )(5𝑥 2 + 7) 12) 𝑦 = (2𝑥 5 )(5𝑥 − 8) First factor Second Factor Derivative Derivative 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 𝑎𝑛𝑠𝑤𝑒𝑟: 𝑦 ′ = 60𝑥 5 − 80𝑥 4 = 20𝑥 4 (3𝑥 − 4) #13-20: Use the quotient rule to find the derivative of the following. 13) 𝑓(𝑥 ) = 14) 𝑔(𝑥 ) = 6 5𝑥+1 4 3𝑥+11 Denominator Numerator Derivative Derivative 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 Create a fraction. Place the expressions in the numerator with a subtraction between. Place the square of the denominator in the denominator −12 𝑎𝑛𝑠𝑤𝑒𝑟: 𝑔′ (𝑥) = (3𝑥+11)2 15) 𝑦 = 16) 𝑦 = 9𝑥 𝑥−5 12𝑥 5𝑥−6 Denominator Numerator Derivative Derivative 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 Create a fraction. Place the expressions in the numerator with a subtraction between. Place the square of the denominator in the denominator −72 𝑎𝑛𝑠𝑤𝑒𝑟: 𝑦 ′ = (5𝑥−6)2 17) 𝑦 = 18) 𝑦 = 3𝑡+1 2𝑡+5 2𝑡+3 4𝑡+5 Denominator Numerator Derivative Derivative Type equation here. 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 Create a fraction. Place the expressions in the numerator with a subtraction between. Place the square of the denominator in the denominator 𝑎𝑛𝑠𝑤𝑒𝑟: 𝑑𝑦 𝑑𝑥 −2 = (4𝑡+5)2 19) 𝑔(𝑥 ) = 20) 𝑔(𝑥 ) = 𝑥2 𝑥−4 𝑥2 𝑥−2 Denominator Numerator Derivative 2 Derivative 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡𝑜𝑝 𝑑𝑜𝑤𝑛 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡𝑡𝑜𝑚 𝑢𝑝 Create a fraction. Place the expressions in the numerator with a subtraction between. Place the square of the denominator in the denominator 𝑎𝑛𝑠𝑤𝑒𝑟: 𝑔′ (𝑥) = 𝑥(𝑥−4) (𝑥−2)2 #21-26: a) Find the slope of the tangent line to the graph of the function for the given value of x (or t). b) Find the equation of the tangent line to the graph of the function for the given value of x (or t). 21) 𝑦 = (2𝑥 + 3)(3𝑥 – 4); 𝑥 = 2 22) 𝑦 = (3𝑥 – 4)(5𝑥 – 8); x = 3 (derivative computed in #1 / 2) 22a) 𝑚 = 46 22b) 𝑦 = 46𝑥 − 103 #21-26: a) Find the slope of the tangent line to the graph of the function for the given value of x (or t). b) Find the equation of the tangent line to the graph of the function for the given value of x (or t). 23) 𝑔(𝑡) = (2𝑡 – 1)(3𝑡 + 5) ;t = 4 24) 𝑔(𝑡) = (3𝑡 2 + 5𝑡)(2𝑡 + 1); 𝑡 = −2 (derivative computed in 7 / 8) 𝑎𝑛𝑠𝑤𝑒𝑟 24𝑎) 𝑚 = 25 24𝑏) 𝑦 = 25𝑡 + 44 #21-26: a) Find the slope of the tangent line to the graph of the function for the given value of x (or t). b) Find the equation of the tangent line to the graph of the function for the given value of x (or t). 25) 𝑓(𝑥 ) = 26) 𝑔(𝑥 ) = 6 5𝑥+1 ;𝑥= 1 4 3𝑥+11 ; 𝑥 = −3 (derivative computed in 13 / 14) 𝑎𝑛𝑠𝑤𝑒𝑟 26𝑎) 𝑚 = −3 26𝑏) 𝑦 = −3𝑥 − 7