Derivatives of Products and Quotients Lesson 4.2.

Slides:

Advertisements

Similar presentations

Product & Quotient Rules Higher Order Derivatives

Advertisements

Product & Quotient Rules Higher Order Derivatives Lesson 3.3.

The Quotient Rule Brought To You By Tutorial Services The Math Center.

Do Now Find the tangents to the curve at the points where

Chapter 3 The Derivative Definition, Interpretations, and Rules.

Business Calculus Derivatives. 1.5 – 1.7 Derivative Rules  Linear Rule:   Constant Rule:   Power Rule:   Coefficient Rule:   Sum/Difference Rule:

Section 2.3: Product and Quotient Rule. Objective: Students will be able to use the product and quotient rule to take the derivative of differentiable.

Polynomial and Rational Functions

Section 12.1 Techniques for Finding Derivative. Constant Rule Power Rule Sum and Difference Rule.

Product and Quotient Rules and Higher – Order Derivatives

Evaluating Limits Analytically Lesson What Is the Squeeze Theorem? Today we look at various properties of limits, including the Squeeze Theorem.

Antiderivatives Lesson 7.1B Down with Derivatives.

Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a x Let a = 2 Graph the function and it's derivative.

Differentiation Formulas

3.3 Techniques of Differentiation Derivative of a Constant (page 191) The derivative of a constant function is 0.

If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: The derivative of a constant is zero.

Chapter 3: Derivatives Section 3.3: Rules for Differentiation

Slide 3- 1 Rule 1 Derivative of a Constant Function.

4-4: The Fundamental Theorems Definition: If f is continuous on [ a,b ] and F is an antiderivative of f on [ a,b ], then: The Fundamental Theorem:

3.3 Rules for Differentiation Quick Review In Exercises 1 – 6, write the expression as a power of x.

Differentiation Formulas

3.3 Rules for Differentiation Positive Integer Powers, Multiples, Sums and Differences Products and Quotients Negative Integer Powers of x Second and Higher.

h Let f be a function such that lim f(2 + h) - f(2) = 5.

4.2:Derivatives of Products and Quotients Objectives: Students will be able to… Use and apply the product and quotient rule for differentiation.

1.6 – Differentiation Techniques The Product and Quotient Rules.

The Fundamental Theorems of Calculus Lesson 5.4. First Fundamental Theorem of Calculus Given f is  continuous on interval [a, b]  F is any function.

Lesson 4-2 Operations on Functions. We can do some basic operations on functions.

Product and Quotient Rules. Product Rule Many people are tempted to say that the derivative of the product is equal to the product of the derivatives.

Product and Quotient Rule By: Jenna Neil Lori Hissick And Evan Zimmerman Period 4.

1 §3.2 Some Differentiation Formulas The student will learn about derivatives of constants, the product rule,notation, of constants, powers,of constants,

Antiderivatives Lesson 7.1A. Think About It Suppose this is the graph of the derivative of a function What do we know about the original function? Critical.

1 §2.2 Product and Quotient Rules The student will learn about derivatives marginal averages as used in business and economics, and involving products,involving.

Inverse Trigonometric Functions: Integration

D ERIVATIVES Review- 6 Differentiation Rules. For a function f(x) the instantaneous rate of change along the function is given by: Which is called the.

Quotient Rule Finding the derivative of a function using the Quotient Rule Andrew Conway.

December 6, 2012 AIM : How do we find the derivative of quotients? Do Now: Find the derivatives HW2.3b Pg #7 – 11 odd, 15, 65, 81, 95, 105 –

AP Calculus Chapter 3 Section 3

DO NOW: Write each expression as a sum of powers of x:

1 Basic Differentiation Rules Lesson 3.2A. 2 Basic Derivatives Constant function – Given f(x) = k Then f’(x) = 0 Power Function –Given f(x) = x n Then.

1. 2 Translating Addition Phrases  The sum of 6 and a number  8 more than a number  A number plus 5  A number increased by 7.

2-3: Product / Quotient Rules & Other Derivatives ©2002 Roy L. Gover Objectives: Learn and use the product & quotient rules. Derive.

Calculus I Ms. Plata Fortunately, several rules have been developed for finding derivatives without having to use the definition directly. Why?

The Product and Quotient Rules and Higher-Order Derivatives Calculus 2.3.

Calculus Section 2.3 The Product and Quotient Rules and Higher-Order Derivatives.

Patel – Honors Classes Only Page 243 # Factoring Polynomials 2/6/14 Thursday.

AP CALCULUS 1008 : Product and Quotient Rules. PRODUCT RULE FOR DERIVATIVES Product Rule: (In Words) ________________________________________________.

4.2:DERIVATIVES OF PRODUCTS AND QUOTIENTS Objectives: To use and apply the product and quotient rule for differentiation.

Calculating the Derivative

The Product and Quotient rules

Review- 6 Differentiation Rules

Warm-Up Find the derivative.

3.1 Polynomial & Exponential Derivatives

Rational functions are quotients of polynomial functions.

Warm up: Below is a graph of f(x). Sketch the graph of f’(x)

The Mean Value Theorem for Integrals – Average Value

Derivatives of Products and Quotients

Product and Quotient Rules

Techniques for Finding Derivatives

Definition of the Derivative

Representation of Functions by Power Series

2.4 Product and Quotient Rules

Evaluating Limits Analytically

Antiderivatives Lesson 7.1A.

Rational Functions Lesson 9.4.

Down with Derivatives Antiderivatives Lesson 7.1B.

PRODUCT AND QUOTIENT RULES AND HIGHER-ORDER DERIVATIVES

Limits Lesson 3.1.

Review Mathematics Skills

UNIT IV Derivatives of Products and Quotients

PRODUCT AND QUOTIENT RULES AND HIGHER-ORDER DERIVATIVES

Presentation transcript:

Derivatives of Products and Quotients Lesson 4.2

Quotients Rule! 2 Products Rule! This lesson will show us how to take derivatives of products and quotients.

Review We know what to do with constant times a function For k f(x) We also know what to do with the sum of functions When Then 3

Product Rule Consider the product of two functions It can be shown (see proof, pg 215) that 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 5

Quotient Rule When our function is the quotient of two other functions … The quotient rule specifies the derivative 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

OK … Try That! Use the quotient rule on the following functions 7

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

Average Cost Suppose the cost for manufacturing x items is 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 9

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