Used for composite functions

Slides:

Advertisements

Similar presentations

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

Advertisements

U2 L8 Chain and Quotient Rule CHAIN & QUOTIENT RULE

The Chain Rule Section 3.6c.

Equations of Tangent Lines

Find the slope of the tangent line to the graph of f at the point ( - 1, 10 ). f ( x ) = 6 - 4x

Point Value : 20 Time limit : 2 min #1 Find. #1 Point Value : 30 Time limit : 2.5 min #2 Find.

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.

2.5 Implicit Differentiation. Implicit and Explicit Functions Explicit FunctionImplicit Function But what if you have a function like this…. To differentiate:

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.

The Power Rule  If we are given a power function:  Then, we can find its derivative using the following shortcut rule, called the POWER RULE:

2.4 Chain Rule. Chain Rule If y=f(u) is a differentiable function of u and u=g(x) is a differentiable function of x then y=f(g(x)) is a differentiable.

Section 2.4 – The Chain Rule. Example 1 If and, find. COMPOSITION OF FUNCTIONS.

Section 2.4 – The Chain Rule

Section 6.1: Euler’s Method. Local Linearity and Differential Equations Slope at (2,0): Tangent line at (2,0): Not a good approximation. Consider smaller.

Differentiating exponential functions.

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

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

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.

In this section, we will investigate a new technique for finding derivatives of curves that are not necessarily functions.

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

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.

3.6The Chain Rule. We now have a pretty good list of “shortcuts” to find derivatives of simple functions. Of course, many of the functions that.

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.

Objectives: 1. Be able to find the derivative of function by applying the Chain Rule Critical Vocabulary: Chain Rule Warm Ups: 1.Find the derivative of.

Slide 3- 1 Quick Quiz Sections 3.4 – Implicit Differentiation.

Derivatives Test Review Calculus. What is the limit equation used to calculate the derivative of a function?

Further Differentiation and Integration

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.

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.

You can do it!!! 2.5 Implicit Differentiation. How would you find the derivative in the equation x 2 – 2y 3 + 4y = 2 where it is very difficult to express.

The Chain Rule Composite Functions When a function is composed of an inner function and an outer function, it is called a “composite function” When a.

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

The Chain Rule. The Chain Rule Case I z x y t t start with z z is a function of x and y x and y are functions of t Put the appropriate derivatives along.

Derivative Shortcuts -Power Rule -Product Rule -Quotient Rule -Chain Rule.

After the test… No calculator 3. Given the function defined by for a) State whether the function is even or odd. Justify. b) Find f’(x) c) Write an equation.

Chapter 3.6 Chain Rule. Objectives Differentiate composite functions using the Chain Rule. Find the slopes of parameterized curves.

UNIT 2 LESSON 9 IMPLICIT DIFFERENTIATION 1. 2 So far, we have been differentiating expressions of the form y = f(x), where y is written explicitly in.

DIFFERENCE QUOTIENT, DERIVATIVE RULES, AND TANGENT LINES RIZZI – CALC BC.

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.

§ 4.2 The Exponential Function e x.

Chapter 3 Derivatives.

Implicit Differentiation

Implicit Differentiation

Section 3.7 Implicit Functions

Find the equation of the tangent line for y = x2 + 6 at x = 3

MTH1170 Implicit Differentiation

3.6 Chain Rule.

Calculus Section 3.6 Use the Chain Rule to differentiate functions

Techniques of Differentiation

The Derivative and the Tangent Line Problems

Unit 6 – Fundamentals of Calculus Section 6

2.4 The Chain Rule.

Implicit Differentiation

Exam2: Differentiation

Implicit Differentiation

Differentiate. f (x) = x 3e x

Derivatives: definition and derivatives of various functions

COMPOSITION OF FUNCTIONS

The Chain Rule Section 3.4.

2.5 Implicit Differentiation

The Chain Rule Section 3.6b.

The Chain Rule Section 2.4.

More with Rules for Differentiation

Chapter 4 More Derivatives Section 4.1 Chain Rule.

2.5 Basic Differentiation Properties

Presentation transcript:

Used for composite functions The Chain Rule Used for composite functions

How do we . . . Find the derivative of a function with the following format:

This is a composite function It can be thought of as being composed of two functions:

To differentiate: The following scheme can be used:

In our example:

A few examples: Find the derivatives:

Taking it further: Find the equation of the tangent line at . We need to find an equation in the form The slope will be found using the derivative.

Taking it further: Find the equation of the tangent line at .

Find a point on the line. Find the equation of the tangent line at . Point on line: y-value

Taking it further: Find the equation of the tangent line at . Slope of line – take derivative:

Find the equation. Format: