# Derivatives of Products and Quotients UNIT IV Derivatives of Products and Quotients Dr. T. VENKATESAN Assistant Professor Department of Statistics St. Josephs College, Trichy-2. Quotients Products Rule! This Thislesson lessonwill willshow showus

ushow howtototake take derivatives derivativesofofproducts productsand andquotients. quotients. 2 Review We know what to do with constant times a function For k f(x) Dx k f ( x) k f '( x)

We also know what to do with the sum of functions When f ( x) h( x) k ( x) Then f '( x) h '( x) k '( x) 3 Product Rule Consider the product of two functions f ( x) h( x) k ( x)

It can be shown (see proof, pg 215) that f '( x) h( x) k '( x) k ( x) h '( x) In words: The first function times the derivative of the second plus the second times the derivative of the first 4 Try It Out Given the following functions which are products Determine the derivatives y 5 x 2 1 4 x 3

f ( x) 2 x 3 x1 5 Quotient Rule When our function is the quotient of two other functions p( x) f ( x) q( x)

The quotient rule specifies the derivative f '( x) q( x) p '( x) p ( x) q '( x) q ( x) 2 In words: 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

6 A Memory Trick hi ( x) Given f ( x) ho( x) Then f '( x) ho( x) Dx hi ( x) hi ( x) Dx ho( x) ho( x) 2 OK Try That!

Use the quotient rule on the following functions 6 x 11 y 8x 1 x2 4x y x 3 h( z ) z z 2.2

3.2 5 8 Average Cost Suppose we have a function y = C(x) which gives us the cost of manufacturing x items The average cost is C ( x) C ( x) x Then the marginal average cost is

C '( x) 9 Average Cost Suppose the cost for manufacturing x items is 3x 2 C ( x) x4 Write the function for the average cost What is the marginal average cost? Determine rate of change of the average cost for 5 items for 50 items

10 Assignment Lesson 4.2A Page 259 Exercises 1 33 odd Lesson 4.2B Page 260 Exercises 39 49 odd 11