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

Slides:

Advertisements

Similar presentations

An introduction to calculus…

Advertisements

8.2 Integration by parts.

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

The Derivative. Definition Example (1) Find the derivative of f(x) = 4 at any point x.

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

3.6 Derivatives of Logarithmic Functions 1Section 3.6 Derivatives of Log Functions.

Calculus and Analytic Geometry II Cloud County Community College Spring, 2011 Instructor: Timothy L. Warkentin.

The Power Rule and other Rules for Differentiation Mr. Miehl

Divisibility rules.

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.

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

The Derivative Definition, Interpretations, and Rules.

2.6 – Limits involving Infinity, Asymptotes

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

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

Section 6.1 Polynomial Derivatives, Product Rule, Quotient Rule.

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.

The Quotient Rule. The following are examples of quotients: (a) (b) (c) (d) (c) can be divided out to form a simple function as there is a single polynomial.

1.6 – Differentiation Techniques The Product and Quotient Rules.

DIFFERENTIATING “COMBINED” FUNCTIONS PRODUCTS. PRODUCTS: A GAMBIER EXAMPLE LARGE RATS!!!!

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) ________________________________________________.

Differentiating “Combined” Functions Deriving Product Rule for Differentiation.

5.3 Definite Integrals and Antiderivatives Objective: SWBAT apply rules for definite integrals and find the average value over a closed interval.

CHAPTER Continuity The Product and Quotient Rules Though the derivative of the sum of two functions is the the sum of their derivatives, an analogous.

Math 1304 Calculus I 2.3 – Rules for Limits.

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

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.

Derivatives of Logarithmic Functions Objective: Obtain derivative formulas for logs.

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

In this section, we will learn about: Differentiating composite functions using the Chain Rule. DERIVATIVES 3.5 The Chain Rule.

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.

Product and Quotient Rule Find the derivative of the function using the Product Rule Find the derivative of the function using the Quotient Rule Find the.

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

3.5 – Implicit Differentiation

Differentiate:What rule are you using?. Aims: To learn results of trig differentials for cos x, sin x & tan x. To be able to solve trig differentials.

Lecture 5 Difference Quotients and Derivatives. f ‘ (a) = slope of tangent at (a, f(a)) Should be “best approximating line to the graph at the point (a,f(a))”

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,

Basic Derivatives Brought To You By: Tutorial Services The Math Center.

Calculus, Section 1.3.

Rules for Differentiation

MTH1150 Rules of Differentiation

Product and Quotient Rules

Derivatives of Logarithmic Functions

Section 14.2 Computing Partial Derivatives Algebraically

Math 1304 Calculus I 2.3 – Rules for Limits.

Sullivan Algebra and Trigonometry: Section 8.3

Positive x Positive = Positive Positive x Negative = Negative

Chapter 3 Derivatives.

Rules for Differentiation

Derivatives of Logarithmic Functions

6.3 Trigonometric Identities

Differentiating “Combined” Functions ---Part I

Sec 3.6: DERIVATIVES OF LOGARITHMIC FUNCTIONS

1 Introduction to Algebra: Integers.

Differentiating “Combined” Functions ---Part I

Chapter 2 Differentiation.

all slides © Christine Crisp

Limits, continuity, and differentiability computing derivatives

Differentiation Rules and formulas

31 – Power, Product, Quotient Rule No Calculator

Rules for Differentiation

6.3 Trigonometric Identities

Combinations of Functions

Table of Contents 4. Section 2.3 Basic Limit Laws.

Presentation transcript:

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

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 product) of these functions without going back to the difference quotient?

The Limit of the Sum of Two Functions

The Limit of the Product of Two Functions

In the course of the calculation above, we said that Is this actually true? Is it ALWAYS true? Continuity Required! xx+h

The Limit of the Product of Two Functions Would be zero if g were continuous at x=a. Is it?

The Limit of the Product of Two Functions g is continuous at x=a because g is differentiable at x=a.