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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.9.73
Chapter 3, Problem 3.9.73

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In (2x - 1)(x + 2)³ / (1 - 4x)²

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First, recognize that the function f(x) = ln((2x - 1)(x + 2)^3 / (1 - 4x)^2) is a natural logarithm of a quotient. Use the logarithmic property ln(a/b) = ln(a) - ln(b) to separate the terms.
Apply the property of logarithms ln(a * b) = ln(a) + ln(b) to further break down the expression: ln((2x - 1)(x + 2)^3) - ln((1 - 4x)^2).
Use the power rule for logarithms, ln(a^b) = b * ln(a), to simplify the expression: ln(2x - 1) + 3 * ln(x + 2) - 2 * ln(1 - 4x).
Differentiate each term separately using the chain rule. For ln(2x - 1), the derivative is 1/(2x - 1) * 2. For 3 * ln(x + 2), the derivative is 3 * (1/(x + 2)). For -2 * ln(1 - 4x), the derivative is -2 * (-4/(1 - 4x)).
Combine the derivatives from each term to find f'(x): f'(x) = 2/(2x - 1) + 3/(x + 2) + 8/(1 - 4x).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the function's graph at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.
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Derivatives

Logarithmic Properties

Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log(a*b) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^b) = b*log(a)). These properties are particularly useful in calculus for simplifying complex functions before differentiation.
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Change of Base Property

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of two functions u and x (i.e., y = f(u) and u = g(x)), then the derivative of y with respect to x is the product of the derivative of f with respect to u and the derivative of g with respect to x. This rule is essential for handling functions that are nested within each other.
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Related Practice
Textbook Question

State the derivative rule for the logarithmic function f(x)=log(subscript b)x. How does it differ from the derivative formula for ln x?

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Textbook Question

Complete the following statement. If dy/dx is small, then small changes in x will result in relatively ______ changes in the value of y.

Textbook Question

Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.

Textbook Question

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>

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Textbook Question

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = x^10x

Textbook Question

The bottom of a large theater screen is 3 ft above your eye level and the top of the screen is 10 ft above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of 3 ft/s while looking at the screen. What is the rate of change of the viewing angle θ when you are 30 ft from the wall on which the screen hangs, assuming the floor is horizontal (see figure)? <IMAGE>