Chapter 3 Chain Rule.

Slides:

Advertisements

Similar presentations

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.

Advertisements

By: Kelley Borgard Block 4A

Product and Quotient Rules and Higher – Order Derivatives

Warm Up: h(x) is a composite function of f(x) and g(x). Find f(x) and g(x)

CHAPTER Continuity Integration by Parts The formula for integration by parts  f (x) g’(x) dx = f (x) g(x) -  g(x) f’(x) dx. Substitution Rule that.

MATH 31 LESSONS Chapters 6 & 7: Trigonometry 9. Derivatives of Other Trig Functions.

Topic 8 The Chain Rule By: Kelley Borgard Block 4A.

In this section, we will learn about: Differentiating composite functions using the Chain Rule. DIFFERENTIATION RULES 3.4 The Chain Rule.

3.6 The Chain Rule We know how to differentiate sinx and x² - 4, but how do we differentiate a composite like sin (x² - 4)? –The answer is the Chain Rule.

2.4 The Chain Rule Remember the composition of two functions? The chain rule is used when you have the composition of two functions.

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

Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 3 Integration.

2-1 The Derivative and the Tangent Line Problem 2-2 Basic Differentiation Rules and Rates of Change 2-3 Product/Quotient Rule and Higher-Order Derivatives.

Section 3.4 The Chain Rule. Consider the function –We can “decompose” this function into two functions we know how to take the derivative of –For example.

 We now know how to differentiate sin x and x 2, but how do we differentiate a composite function, such as sin (x 2 -4)?

HIGHER ORDER DERIVATIVES Product & Quotient Rule.

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?

Examples A: Derivatives Involving Algebraic Functions.

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.

1 The Chain Rule Section After this lesson, you should be able to: Find the derivative of a composite function using the Chain Rule. Find the derivative.

Ch. 4 – More Derivatives 4.1 – Chain Rule. Ex: Find the derivative of h(x) = sin(2x 2 ). –This is not a product rule problem because it has one function.

WARM UP: h(x)=f(g(x)). Identify f(x) and g(x). 1. h(x)=sin(2x) 2. h(x)=(x 2 +2) 1\2 3. If h(x)=f(g(j(x))), identify f(x), g(x) and j(x): h(x)=cos 2 (sinx)

Further Differentiation and Integration

Chain Rule 3.5. Consider: These can all be thought of as composite functions F(g(x)) Derivatives of Composite functions are found using the chain rule.

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 –

CHAPTER 4 DIFFERENTIATION NHAA/IMK/UNIMAP. INTRODUCTION Differentiation – Process of finding the derivative of a function. Notation NHAA/IMK/UNIMAP.

2.4: THE CHAIN RULE. Review: Think About it!!  What is a derivative???

The Product and Quotient Rules for Differentiation.

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

Power Rule is a corallary to Chain Rule. Power Rule If f(x) = x n then f ' (x) = n x (n-1) Replacing x by g(x) gives.

3.1 The Product and Quotient Rules & 3.2 The Chain Rule and the General Power Rule.

2-1 The Derivative and the Tangent Line Problem 2-2 Basic Differentiation Rules and Rates of Change 2-3 Product/Quotient Rule and Higher-Order Derivatives.

Friday, February 12, 2016MAT 145. Friday, February 12, 2016MAT 145.

1. Find the derivatives of the functions sin x, cos x, tan x, sec x, cot x and cosec x. 2. Find the derivatives of the functions sin u, cos u, tan u,

Monday, February 15, 2016MAT 145. Function typeDerivative Rule Constant for constant c Power for any real number n Product of constant and functions for.

Lesson 3-7 Higher Order Deriviatives. Objectives Find second and higher order derivatives using all previously learned rules for differentiation.

Inverse trigonometric functions and their derivatives

Chapter 3 Derivatives.

Chapter 3 Derivatives Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Differentiation Product Rule By Mr Porter.

Section 3.3 The Product and Quotient Rule

Find the derivative Find the second derivative

Calculus Section 3.6 Use the Chain Rule to differentiate functions

CHAPTER 4 DIFFERENTIATION.

Derivatives-Second Part

Copyright © Cengage Learning. All rights reserved.

Chapter 3 Derivatives.

Product and Quotient Rules

Chain Rule AP Calculus.

2.4 The Chain Rule.

2.4 The Chain Rule Use the Chain Rule to find derivative of a composite function. Use the General Power Rule to find derivative of a function. Simplify.

Chapter 3 Integration Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Product and Quotient Rules and Higher Order Derivatives

Exam2: Differentiation

The Chain Rule Section 4 Notes.

Trigonometric functions

Continuity and Differentiation

Exam2: Differentiation

Section 2.3 Day 1 Product & Quotient Rules & Higher order Derivatives

The Chain Rule Section 3.4.

Used for composite functions (a function within a function)…

6-1: Operations on Functions (+ – x ÷)

3.5 Chain Rule.

Sec 3.3: Derivatives Of Trigonometric Functions

Find the derivative of the following function: y(x) = -3 sec 9 x.

The Chain Rule Section 2.4.

Calculus 3-7 CHAIN RULE.

Chapter 3 Derivatives.

Chapter 4 More Derivatives Section 4.1 Chain Rule.

Differentiation Rules for Products, Quotients,

Presentation transcript:

Chapter 3 Chain Rule

Chain rule Derivative of outside function times derivative of the inside function

The Chain Rule Remember the composition of two functions? The chain rule is used when you have the composition of two functions.

f(x) = x3 Find y’ for y = (x2 + 1)3 g(x) = x2 + 1 y’ = 3(x2 + 1)2 (2x) = 6x(x2 + 1)2 Find y’ for y = (3x – 2x2)3 y’ = 3(3x – 2x2)2(3 – 4x) Find f’(x) for

Differentiate rewritten as = -7(2t – 3)-2 g’(t) = 14(2t – 3)-3(2) Differentiate rewritten as Factor

Differentiate rewritten as Bot * Top’ – Top * Bot’ (Bot)2 Quotient Rule

Differentiate

Derivatives of Trigonometric Functions

Applying the Chain Rule to trigonometric functions y = sin 2x y = cos (x – 1) y = tan 3x y = cos (3x2) y = cos2 3x y’ = (cos 2x) (2) = 2 cos 2x y’ = -sin (x – 1) (1) = -sin (x – 1) y’ = sec2 3x (3) = 3 sec2 3x y’ = -sin (3x2) (6x) = -6x sin (3x2) rewritten as y = (cos 3x)2 y’ = 2(cos 3x)1 (-sin 3x) (3) y’ = -6 cos 3x sin 3x

Differentiate rewritten as