§ 4.2 The Exponential Function e x.

Slides:

Advertisements

Similar presentations

Product & Quotient Rules Higher Order Derivatives

Advertisements

Unit 6 – Fundamentals of Calculus Section 6

The Chain Rule Section 3.6c.

3.2 Inverse Functions and Logarithms 3.3 Derivatives of Logarithmic and Exponential functions.

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.

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

Section 3.5a. Evaluate the limits: Graphically Graphin What happens when you trace close to x = 0? Tabular Support Use TblStart = 0, Tbl = 0.01 What does.

The Derivative. Objectives Students will be able to Use the “Newton’s Quotient and limits” process to calculate the derivative of a function. Determine.

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.

Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 55 § 4.5 The Derivative of ln x.

Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 55 § 4.2 The Exponential Function e x.

Section 2.3: Product and Quotient Rule. Objective: Students will be able to use the product and quotient rule to take the derivative of differentiable.

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.

Equations of Tangent Lines April 21 st & 22nd. Tangents to Curves.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 1 of 55 Chapter 4 The Exponential and Natural Logarithm.

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

How can one use the derivative to find the location of any horizontal tangent lines? How can one use the derivative to write an equation of a tangent line.

The chain rule (2.4) October 23rd, I. the chain rule Thm. 2.10: The Chain Rule: If y = f(u) is a differentiable function of u and u = g(x) is a.

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

2.3 The Product and Quotient Rules and Higher Order Derivatives

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

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

Section 6.1 Polynomial Derivatives, Product Rule, Quotient Rule.

1.6 – Tangent Lines and Slopes Slope of Secant Line Slope of Tangent Line Equation of Tangent Line Equation of Normal Line Slope of Tangent =

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

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.3 Product and Quotient Rules for Differentiation.

5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover ( Review properties of natural logarithms Differentiate natural logarithm.

A) b) c) d) Solving LOG Equations and Inequalities **SIMPLIFY all LOG Expressions** CASE #1: LOG on one side and VALUE on other Side Apply Exponential.

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.

Aim: How do we find the derivative by limit process? Do Now: Find the slope of the secant line in terms of x and h. y x (x, f(x)) (x + h, f(x + h)) h.

Calculus and Analytical Geometry

2.1 The Derivative and The Tangent Line Problem Slope of a Tangent Line.

Warm Up. Equations of Tangent Lines September 10 th, 2015.

Problem of the Day - Calculator Let f be the function given by f(x) = 2e4x. For what value of x is the slope of the line tangent to the graph of f at (x,

Basic Rules of Derivatives Examples with Communicators Calculus.

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

Chapter 4 Logarithm Functions

Implicit Differentiation

2-4 Rates of change & tangent lines

§ 1.3 The Derivative.

Implicit Differentiation

(8.2) - The Derivative of the Natural Logarithmic Function

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

Objectives for Section 11.3 Derivatives of Products and Quotients

Chapter 11 Additional Derivative Topics

3.1 Polynomial & Exponential Derivatives

Derivative of an Exponential

§ 4.5 The Derivative of ln x.

§ 4.4 The Natural Logarithm Function.

Techniques of Differentiation

The Derivative and the Tangent Line Problems

Unit 6 – Fundamentals of Calculus Section 6

Implicit Differentiation

Calculating the Derivative

Solve the equation for x. {image}

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

Derivatives of Logarithmic Functions

(This is the slope of our tangent line…)

The Product and Quotient Rules

Exponential and Logarithmic Derivatives

§ 4.6 Properties of the Natural Logarithm Function.

§ 4.3 Differentiation of Exponential Functions.

Tutorial 5 Logarithm & Exponential Functions and Applications

The Chain Rule Section 3.6b.

Section 2.5 Day 1 AP Calculus.

More with Rules for Differentiation

Presentation transcript:

§ 4.2 The Exponential Function e x

Section Outline e The Derivatives of 2x, bx, and ex

The Number e Definition Example e: An irrational number, approximately equal to 2.718281828, such that the function f (x) = bx has a slope of 1, at x = 0, when b = e

The Derivative of 2x

Solving Exponential Equations EXAMPLE Calculate. SOLUTION

The Derivatives of bx and ex

Solving Exponential Equations EXAMPLE Find the equation of the tangent line to the curve at (0, 1). SOLUTION We must first find the derivative function and then find the value of the derivative at (0, 1). Then we can use the point-slope form of a line to find the desired tangent line equation. This is the given function. Differentiate. Use the quotient rule.

Solving Exponential Equations CONTINUED Simplify. Factor. Simplify the numerator. Now we evaluate the derivative at x = 0.

Solving Exponential Equations CONTINUED Now we know a point on the tangent line, (0, 1), and the slope of that line, -1. We will now use the point-slope form of a line to determine the equation of the desired tangent line. This is the point-slope form of a line. (x1, y1) = (0, 1) and m = -1. Simplify.