2.3 The Product Rule Notice the order doesn’t matter!

Slides:

Advertisements

Similar presentations

2.5 Implicit Differentiation Niagara Falls, NY & Canada Photo by Vickie Kelly, 2003.

Advertisements

The Chain Rule Section 3.6c.

C1: Tangents and Normals

The derivative as the slope of the tangent line (at a point)

Derivative Review Part 1 3.3,3.5,3.6,3.8,3.9. Find the derivative of the function p. 181 #1.

DIFFERENTIATION & INTEGRATION CHAPTER 4.  Differentiation is the process of finding the derivative of a function.  Derivative of INTRODUCTION TO DIFFERENTIATION.

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.

Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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

CHAPTER Continuity CHAPTER Derivatives of Polynomials and Exponential Functions.

In this section, we will begin investigating some more advanced techniques for integration – specifically, integration by parts.

First Order Linear Equations Integrating Factors.

3.9: Derivatives of Exponential and Log Functions Objective: To find and apply the derivatives of exponential and logarithmic functions.

Derivatives of Logarithmic Functions

3.8: Derivatives of inverse trig functions

Special Derivatives. Derivatives of the remaining trig functions can be determined the same way. 

7.3* The Natural Exponential Function INVERSE FUNCTIONS In this section, we will learn about: The natural exponential function and its properties.

Differentiation Formulas

3.3 Rules for Differentiation. What you’ll learn about Positive Integer Powers, Multiples, Sums and Differences Products and Quotients Negative Integer.

Chapter 3: Derivatives Section 3.3: Rules for Differentiation

MAT 125 – Applied Calculus 3.2 – The Product and Quotient Rules.

3.3: Rules of Differentiation Objective: Students will be able to… Apply the Power Rule, Sum and Difference Rule, Quotient and Product Rule for differentiation.

1 Implicit Differentiation. 2 Introduction Consider an equation involving both x and y: This equation implicitly defines a function in x It could be defined.

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

Chapter3: Differentiation DERIVATIVES OF TRIGONOMETRIC FUNCTIONS: Chain Rule: Implicit diff. Derivative Product Rule Derivative Quotient RuleDerivative.

Chapter 4 The Exponential and Natural Logarithm Functions.

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.

GOAL: USE DEFINITION OF DERIVATIVE TO FIND SLOPE, RATE OF CHANGE, INSTANTANEOUS VELOCITY AT A POINT. 3.1 Definition of Derivative.

3.3 Rules for Differentiation Colorado National Monument.

3.7 – Implicit Differentiation An Implicit function is one where the variable “y” can not be easily solved for in terms of only “x”. Examples:

4.1 : Anti-derivatives Greg Kelly, Hanford High School, Richland, Washington.

SAT Prep. Basic Differentiation Rules and Rates of Change Find the derivative of a function using the Constant Rule Find the derivative of a function.

Some needed trig identities: Trig Derivatives Graph y 1 = sin x and y 2 = nderiv (sin x) What do you notice?

Sec. 3.3: Rules of Differentiation. The following rules allow you to find derivatives without the direct use of the limit definition. The Constant Rule.

More with Rules for Differentiation Warm-Up: Find the derivative of f(x) = 3x 2 – 4x 4 +1.

MAT 1221 Survey of Calculus Section 2.2 Some Rules for Differentiation

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

Math 1411 Chapter 2: Differentiation 2.3 Product and Quotient Rules and Higher-Order Derivatives.

A little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears.

Warm up: Find the derivative of the following. Question: Can we find the derivative of a product by multiplying the two derivatives? Test this with some.

Chapter 9 & 10 Differentiation Learning objectives: 123 DateEvidenceDateEvidenceDateEvidence Understand the term ‘derivative’ and how you can find gradients.

Aim: How do we take second derivatives implicitly? Do Now: Find the slope or equation of the tangent line: 1)3x² - 4y² + y = 9 at (2,1) 2)2x – 5y² = -x.

Rules for Differentiation

Product and Quotient Rules

Derivatives of exponentials and Logarithms

Implicit Differentiation

Section 2-3b The Product Rule

Implicit Differentiation

Chapter 3 Differentiation

Used for composite functions

Packet #10 Proving Derivatives

4.1 : Anti-derivatives Greg Kelly, Hanford High School, Richland, Washington.

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

The Derivative and the Tangent Line Problems

Literacy in Maths Gradient Tangent Normal

Exam2: Differentiation

Derivatives of Polynomials and Exponential Functions

Derivative of Logarithm Function

(This is the slope of our tangent line…)

Implicit Differentiation

Equation Review Given in class 10/4/13.

Differentiate. f (x) = x 3e x

Raffle Game Chapter Review.

Exam2: Differentiation

2.5 Implicit Differentiation

The derivative as the slope of the tangent line

Equation Review.

Raffle Game Chapter Review.

More with Rules for Differentiation

Presentation transcript:

2.3 The Product Rule Notice the order doesn’t matter! If ℎ 𝑥 =𝑓 𝑥 𝑔(𝑥), then ℎ ′ 𝑥 = 𝑓 ′ 𝑥 𝑔 𝑥 +𝑓 𝑥 𝑔′(𝑥). ℎ ′ 𝑥 = lim ℎ→0 𝑓 𝑥+ℎ 𝑔(𝑥+ℎ)−𝑓 𝑥 𝑔(𝑥) ℎ Proof: Example: Differentiate 𝑓 𝑥 =( 𝑥 2 −3𝑥)( 𝑥 5 +2) using the product rule. = lim ℎ→0 𝑓 𝑥+ℎ 𝑔 𝑥+ℎ −𝑓 𝑥 𝑔 𝑥+ℎ +𝑓 𝑥 𝑔(𝑥+ℎ)−𝑓 𝑥 𝑔(𝑥) ℎ = lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 𝑔 𝑥+ℎ +𝑓 𝑥 [𝑔(𝑥+ℎ)−𝑔(𝑥]) ℎ = lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ 𝑔 𝑥+ℎ +𝑓 𝑥 𝑔(𝑥+ℎ)−𝑔(𝑥) ℎ = lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ lim ℎ→0 𝑔 𝑥+ℎ + lim ℎ→0 𝑓 𝑥 lim ℎ→0 𝑔(𝑥+ℎ)−𝑔(𝑥) ℎ = 𝑓 ′ 𝑥 𝑔 𝑥 +𝑓 𝑥 𝑔′(𝑥).

Practice Find the value ℎ ′ −1 for the function ℎ 𝑥 =(5 𝑥 3 +7 𝑥 2 +3)(2 𝑥 2 +𝑥+6) −𝟖

In summary … We learned another rule to help us determine the derivative more efficiently - THE PRODUCT RULE! QUESTIONS: p.90-91 #1, 2, 3cdf, 6, 7a, 10 (Find the equation of the normal)