2.4 Product and Quotient Rules Calc 10/3/14

Warm-up 2.4 Product and Quotient Rules Welcome future entrepreneurs and economists! What price should we sell this product for? (The demand function comes from our extensive research of past sales and consumer surveys)

2.4 The Product and Quotient Rules 𝑓(𝑥)= 3 𝑥 2 +4 𝑥 −5 𝑓 𝑥 = 4 𝑥 2 −7𝑥+18 𝑥 2

The Product Rule The derivative of the product of two differentiable functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first. 𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 =𝑓 𝑥 𝑔 ′ 𝑥 +𝑔 𝑥 𝑓′(𝑥)

Ex 1. 𝑦= 3 𝑥 2 +4 𝑥 −5 Find the value of the derivative at (1, -28)

Ex 2. 𝑦=(6−3𝑥)(4𝑥+3 𝑥 2 ) Find the value of the derivative when x=2

Ex 3. y=( 2 𝑥 +1)(3𝑥 −1) Find the value at (4, 16.5)

Ex 4. 𝑦=3𝑥(2 𝑥 2 +5𝑥)

𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑔 𝑥 𝑓 ′ 𝑥 −𝑓 𝑥 𝑔 ′ 𝑥 𝑔 𝑥 2 The Quotient Rule The derivative of the quotient of two differentiable functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. 𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑔 𝑥 𝑓 ′ 𝑥 −𝑓 𝑥 𝑔 ′ 𝑥 𝑔 𝑥 2

Ex 5. 𝑦= 𝑥+4 5𝑥−2

𝐸𝑥 6. 𝑦= 3− 2 𝑥 𝑥+4

Ex 7 a) 𝑦= 5 𝑥 4 8 b) 𝑦=− 3 3𝑥−2 𝑥 2 7𝑥

BE CAREFUL Product Rule does NOT mean to multiply the derivatives of each Quotient Rule does NOT mean to divide the derivatives of the numerator and denominator If you can use the constant multiple rule…. Use it! Saves you a lot of time and effort