DIFFERENTIATING “COMBINED” FUNCTIONS ---PART I CONSTANT MULTIPLES, SUMS AND DIFFERENCES.

Slides:

Advertisements

Similar presentations

Warm up Problem If , find .

Advertisements

However, some functions are defined implicitly. Some examples of implicit functions are: x 2 + y 2 = 25 x 3 + y 3 = 6xy.

Chapter 2 Differentiation. Copyright © Houghton Mifflin Company. All rights reserved.2 | 2 Figure 2.1: Tangent Line to a Graph.

3.1 – Simplifying Algebraic Expressions

The Derivative Function (11/12/05) Since we know how to compute (or at least estimate) the derivative (i.e., the instantaneous rate of change) of a given.

Estimating Quotients. Compatible Numbers Estimate: 5,903 6 = 6,000 6= Estimate: 867 9= Numbers that are easy to compute mentally. 1, =100.

CHAPTER Continuity CHAPTER Derivatives of Polynomials and Exponential Functions.

Chapter 3 The Derivative Definition, Interpretations, and Rules.

Simplifying Expressions and Combining Like Terms

The Power Rule and other Rules for Differentiation Mr. Miehl

Rules for Differentiation. Taking the derivative by using the definition is a lot of work. Perhaps there is an easy way to find the derivative.

The Derivative Definition, Interpretations, and Rules.

Tangents and Differentiation

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

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

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.

AP Calculus BC September 9, 2015 Day 7 – The Chain Rule and Implicit Differentiation.

Implicit Differentiation

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

Differentiating “Combined” Functions Deriving the Product Rule for Differentiation.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Rules for Differentiation Section 3.3.

Differentiating “Combined” Functions ---Part I Constant Multiples, Sums and Differences.

Sec 3.3: Differentiation Rules Example: Constant function.

OPERATIONS WITH DERIVATIVES. The derivative of a constant times a function Justification.

AP CALCULUS 1009 : Product and Quotient Rules. PRODUCT RULE FOR DERIVATIVES Product Rule: (In Words) ________________________________________________.

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

1.1 Variables and Expressions Definition of terms: 1 ). A variable is a letter or symbol used to represent a value that can change. 2). A constant is a.

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.

Sec 3.1: DERIVATIVES of Polynomial and Exponential Example: Constant function.

Basic rules of differentiation Recall that all represent the derivative. All rules follow from the definition of the derivative.

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.

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

Techniques of Differentiation. We now have a shortcut to find a derivative of a simple function. You multiply the exponent by any coefficient in front.

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

Chapter 2 Differentiation. Copyright © Houghton Mifflin Company. All rights reserved.2 | 2 Tangent Line to a Graph.

1-2 Simplifying Expressions. Free powerpoints at

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 1 Quick Review.

Differentiating “Combined” Functions Deriving the Sum and Product Rules for Differentiation.

8.9: Finding Power Series Using Algebra or Calculus Many times a function does not have a simple way to rewrite as the sum of an infinite geometric series.

Basic derivation rules We will generally have to confront not only the functions presented above, but also combinations of these : multiples, sums, products,

Calculus, Section 1.3.

Copyright © Cengage Learning. All rights reserved.

Rules for Differentiation

MTH1150 Rules of Differentiation

Derivatives of Logarithmic Functions

AP Calculus BC September 12, 2016.

Section 14.2 Computing Partial Derivatives Algebraically

2 Chapter Chapter 2 Integers and Introduction to Variables.

Chapter 5 The Definite Integral. Chapter 5 The Definite Integral.

1-2 Simplifying Expressions.

2-4 The Distributive Property

Solve more difficult number problems mentally

AP Calculus Honors Ms. Olifer

Rules for Differentiation

Rules for Differentiation

Sec 3.1: DERIVATIVES of Polynomial and Exponential

Algebraic Expressions, Equations, and Symbols

Techniques Of Differentiation

Writing expression basic

Differentiating “Combined” Functions ---Part I

Sec 3.6: DERIVATIVES OF LOGARITHMIC FUNCTIONS

Differentiating “Combined” Functions ---Part I

Finding Limits Algebraically

Chapter 2 Differentiation.

Differentiation Rules and formulas

Rules for Differentiation

Section 5.2 Definite Integrals

Combinations of Functions

Table of Contents 4. Section 2.3 Basic Limit Laws.

Presentation transcript:

DIFFERENTIATING “COMBINED” FUNCTIONS ---PART I CONSTANT MULTIPLES, SUMS AND DIFFERENCES

ALGEBRAIC COMBINATIONS We have seen that it is fairly easy to compute the derivative of a “simple” function using the definition of the derivative. More complicated functions can be difficult or impossible to differentiate using this method. So we ask... If we know the derivatives of two fairly simple functions, can we deduce the derivative of some algebraic combination (e.g. the sum or difference) of these functions without going back to the difference quotient?

differentiable Note: We are going to assume that all of the functions we talk about in this PowerPoint presentation are differentiable.

THE DERIVATIVE OF A CONSTANT TIMES A FUNCTION Can we do this? Why?

WHAT DOES THIS MEAN?

THE DERIVATIVE OF THE SUM OF TWO FUNCTIONS Can we do this?

THE DERIVATIVE OF THE DIFFERENCE OF TWO FUNCTIONS