3 handouts section3-8
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1. This section discusses how to find derivatives of functions involving logarithms, including ln(u) and logb(u). It outlines the differentiation formulas for these functions and explains logarithmic differentiation. 2. Logarithmic differentiation can be used to differentiate functions that involve products, quotients, or powers, by taking the natural log of both sides, simplifying using properties of logarithms, and then differentiating. 3. Logarithmic differentiation is particularly useful for differentiating functions of the form v/u, where both u and v are functions of the variable. Examples are provided.
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- Composition of functions, where the output of one function becomes the input of another. The order of functions in composition matters.
- Other topics like inverse functions, trigonometric functions, exponentials and logarithms are also reviewed at a high level.
The document
Logarithms- principal properties
Logarithms are the inverse of exponential functions. A logarithm represents the power to which a base number must be raised to equal the output number. Logarithmic functions take the form y=logb(x), where b is the base, x is the output number, and y is the exponent or power. Common bases include base 10 and base e. Logarithmic functions have properties that allow logarithms with the same base to be added, subtracted, and multiplied according to certain rules. These properties can be used to solve exponential equations.
Module 4 exponential and logarithmic functions
This document provides an overview of a module on logarithmic functions. It discusses the definition of logarithmic functions as the inverse of exponential functions, how to graph logarithmic functions by reflecting the graph of the corresponding exponential function across the line y=x, and properties of logarithmic function graphs like their domains, ranges, asymptotes, and behavior. It also covers laws of logarithms and how to solve logarithmic equations. The document is designed to teach students to define logarithmic functions, graph them, use laws of logarithms, and solve simple logarithmic equations.
chapter3.ppt
The document discusses exponential and logarithmic functions. It defines exponential functions as functions of the form f(x) = bx, where b is the base. It provides examples of graphs of exponential functions with different bases. It then introduces logarithmic functions as the inverse of exponential functions, where logarithms are defined as logbx = y if x = by. It provides properties and examples involving logarithmic functions.
Lecture 11 sections 4.3-4.4 logarithmic functions
This document discusses logarithmic functions and their properties. It begins with examples of converting between exponential and logarithmic form. It then covers the natural logarithm function and using transformations of logarithmic functions. The main section discusses properties of logarithms, including the product rule, quotient rule, and power rule for logarithms. It provides examples of writing logarithmic expressions using these rules in both expanded and condensed form.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
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دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهال
The document discusses derivatives of exponential and logarithmic functions. It begins by introducing exponential functions and their role in modeling growth and decay. It then derives the derivative of the exponential function ex as ex and the derivative of the natural logarithm function ln x as 1/x. Finally, it generalizes these results to derivatives of exponential and logarithmic functions with arbitrary bases b > 0 and b ≠ 1.
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3 handouts section3-5
1. The section discusses the chain rule for differentiation, which states that if y is a function of u and u is a function of x, then the derivative of y with respect to x is equal to the derivative of y with respect to u multiplied by the derivative of u with respect to x.
2. It reviews the proper forms of differentiation formulas when combined with the chain rule, including the power rule, and formulas for derivatives of functions involving e^u, a^u, trigonometric functions, and their combinations.
3. Examples are provided to demonstrate the proper application of these differentiation formulas using the chain rule to find derivatives.
4.3 Logarithmic Functions
This document provides an overview of logarithmic functions. It defines logarithms as the inverse of exponents, where the logarithm of a number is the exponent that the base must be raised to to produce that number. It presents examples of solving logarithmic equations by rewriting them as exponential equations. The key properties of logarithms - that they satisfy the same properties as exponents - are described. The document works through multiple examples of evaluating logarithms and using their properties to simplify expressions and solve equations.
Log+Test+Review+Topics.pdf
This document provides a review of topics for logarithmic functions including:
1. Exponent properties, the quadratic formula, and other exponent rules to remember for working with logarithmic expressions.
2. Identifying exponential growth and decay and rules for exponents involving the natural log e.
3. Properties of logarithmic functions like the product, quotient, and power properties as well as how to evaluate, expand, condense, and change the base of logarithmic expressions.
4. How to solve logarithmic equations using the property of equality for logarithms and checking solutions.
Bab 4 dan bab 5 algebra and trigonometry
This document discusses exponential functions and their properties. Exponential functions have the form f(x) = ax, where a is the base. They model phenomena like population growth, compound interest, and radioactive decay. Exponential functions either increase or decrease rapidly depending on whether the base a is greater than or less than 1. Their graphs have a characteristic shape, with the x-axis as a horizontal asymptote. Logarithmic functions are the inverses of exponential functions and can be used to solve equations involving exponential models.
3 handouts section3-4
This document discusses differentiating trigonometric functions. It covers calculating limits of trigonometric expressions, especially using the limit as x approaches 0 of sin(x)/x equals 1. Basic differentiation formulas are provided for sin(x), cos(x), tan(x), cot(x), sec(x), and csc(x). Examples are worked through in class applying these formulas and techniques for limits and derivatives of trigonometric functions.
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Chapter4 exponentialandlogarithmicfunctions-151003150209-lva1-app6891
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Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...
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Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...
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3 handouts section3-8
- 1. Section 3.8 Derivatives of logarithmic functions Learning outcomes After completing this section, you will inshaAllah be able to 1. find derivatives of expressions involving lnu 2. find derivatives of expressions involving logb u 3. explain what is logarithmic differentiation 4. find derivatives using method of logarithmic differentiation 5. differentiate functions of the form v u Recall the following properties which are needed for this section 13.8 • ( )ln ln lnxy x y= + • ln ln ln x x y y ⎛ ⎞ = −⎜ ⎟ ⎝ ⎠ • ( )ln lnr x r x=
- 2. Derivatives of functions involving lnu Derivatives of functions involving logb u 23.8 Differentiation formula for lnu (ln ) 1d u du du u dx = ⋅ Differentiation formula for logb u 1 ln (log ) 1b b d u du du u dx ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = ⋅ How? Using ln log ln b u u b = See examples 1, 2, 3, 4, 5 done in class
- 3. How to perform logarithmic differentiation of f(x)? • We learn more with the help of example. End of 3.8 33.8 See example 6 done in class Main idea • Simplify before differentiating How? • Aim: To differentiate y=f(x) (1) • If f(x) involves products, quotients or powers then o take ‘ln’ on both sides of (1) o simplify using properties of ‘ln’ o differentiate after simplification Suitable when f(x) involves products, quotients or powers Important application of logarithmic differentiation Differentiating functions of the form v u where both u and v are functions of x See examples 7, 8 done in class