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

Slides:

Advertisements

Similar presentations

Variables and Expressions

Advertisements

Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve. 2.3 Derivatives of Trigonometric.

Section 3.3a. The Do Now Find the derivative of Does this make sense graphically???

Section 2.3 – Product and Quotient Rules and Higher-Order Derivatives

U2 L5 Quotient Rule QUOTIENT RULE

The Chain Rule Section 3.6c.

11.3:Derivatives of Products and Quotients

Derivatives - Equation of the Tangent Line Now that we can find the slope of the tangent line of a function at a given point, we need to find the equation.

Dr .Hayk Melikyan Departmen of Mathematics and CS

3.3 Differentiation Formulas

1.4 – Differentiation Using Limits of Difference Quotients

Implicit Differentiation. Objectives Students will be able to Calculate derivative of function defined implicitly. Determine the slope of the tangent.

Chapter 3 The Derivative Definition, Interpretations, and Rules.

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.

The Product Rule The derivative of a product of functions is NOT the product of the derivatives. If f and g are both differentiable, then In other words,

Every slope is a derivative. Velocity = slope of the tangent line to a position vs. time graph Acceleration = slope of the velocity vs. time graph How.

Addition, Subtraction, Multiplication, and Division of Integers

DERIVATIVES 3. If it were always necessary to compute derivatives directly from the definition, as we did in the Section 3.2, then  Such computations.

Differentiation Formulas

Chapter 4 Techniques of Differentiation Sections 4.1, 4.2, and 4.3.

Copyright © Cengage Learning. All rights reserved.

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

Copyright © Cengage Learning. All rights reserved. 2 Derivatives.

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

In this section, we will investigate how to take the derivative of the product or quotient of two functions.

DERIVATIVES 3. If it were always necessary to compute derivatives directly from the definition, as we did in the Section 3.2, then  Such computations.

Warm Up  Brainstorm words that are associated with: SymbolAssociated Word + - x ÷

3.2 The Product and Quotient Rules DIFFERENTIATION RULES In this section, we will learn about: Formulas that enable us to differentiate new functions formed.

3.2 The Product and Quotient Rules DIFFERENTIATION RULES In this section, we will learn about:  Formulas that enable us to differentiate new functions.

3.3 Rules for Differentiation What you’ll learn about Positive integer powers, multiples, sums, and differences Products and Quotients Negative Integer.

Derivatives of Products and Quotients Lesson 4.2.

Differentiation Formulas

The Quotient Rule. Objective  To use the quotient rule for differentiation.  ES: Explicitly assessing information and drawing conclusions.

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

In this section, we will consider the derivative function rather than just at a point. We also begin looking at some of the basic derivative rules.

Chapter 4 Additional Derivative Topics

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.

3.3 Rules for Differentiation Colorado National Monument.

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

Section 3.3 The Product and Quotient Rule. Consider the function –What is its derivative? –What if we rewrite it as a product –Now what is the derivative?

By: Susana Cardona & Demetri Cheatham © Cardona & Cheatham 2011.

SECTION 1.4 EXPONENTS. PRODUCT OF POWERS When you multiply two factors having the same base, keep the common base and add the exponents.

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

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.

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.

Chapter 3 Limits and the Derivative

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 –

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

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.

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

2.3 Basic Differentiation Formulas

A Brief Introduction to Differential Calculus. Recall that the slope is defined as the change in Y divided by the change in X.

3.3 – Rules for Differentiation

§ 4.2 The Exponential Function e x.

0-4/0-5 Operations with Fractions

The Product and Quotient Rules

Rules for Differentiation

3-3 rules for differentiation

Objectives for Section 11.3 Derivatives of Products and Quotients

The Quotient Rule The Quotient Rule is used to find derivatives for functions written as a fraction:

Chapter 11 Additional Derivative Topics

2.3 Basic Differentiation Formulas

Techniques of Differentiation

Quotient Next Slide © Annie Patton.

The Chain Rule Section 3.6b.

2.5 Basic Differentiation Properties

Chapter 3 Additional Derivative Topics

Presentation transcript:

Do Now Find the tangents to the curve at the points where the slope is 4. What is the smallest slope of the curve? At what value of x does the curve have this slope?

Product and Quotient Rules Section 3.3b

Do Now Graphical support??? Find the tangents to the curve at the points where the slope is 4. What is the smallest slope of the curve? At what value of x does the curve have this slope? The derivative: Slope 4, points (1,2) and (–1,–2). Find where the slope is 4: Tangent lines: For smallest slope, minimize The smallest slope is 1, and occurs at x = 0. Graphical support???

We need to derive a new rule for products… As we learned last class, the derivative of the sum of two functions is the sum of their derivatives (and the same holds true for differences of functions). Is there a similar rule for the product of two functions? Let The derivative: However, We need to derive a new rule for products…

Let The derivative: Subtract and add u(x + h)v(x) in the numerator:

Let The derivative:

Rule 5: The Product Rule To find the derivative of a product of two The product of two differentiable functions u and v is differentiable, and To find the derivative of a product of two functions: “The first times the derivative of the second plus the second times the derivative of the first.”

How about when we have a quotient?... The derivative: Subtract and add v(x)u(x) in the numerator:

How about when we have a quotient?... The derivative:

Rule 6: The Quotient Rule At a point where , the quotient of two differentiable functions is differentiable, and To find the derivative of a quotient of two functions: “The bottom times the derivative of the top minus the top times the derivative of the bottom, all divided by the bottom squared.”

Practice Problems Any other method for finding this answer? Find if Let’s use the product rule with and Any other method for finding this answer?

Practice Problems Graphical support: Differentiate Use the quotient rule with and Graphical support:

Practice Problems Let be the product of the functions u and v. Find if From the Product Rule: At our particular point:

Practice Problems Suppose u and v are functions of x that are differentiable at x = 2. Also suppose that Find the values of the following derivatives at x = 2. (a)

Practice Problems Suppose u and v are functions of x that are differentiable at x = 2. Also suppose that Find the values of the following derivatives at x = 2. (b)

Practice Problems Suppose u and v are functions of x that are differentiable at x = 2. Also suppose that Find the values of the following derivatives at x = 2. (c)