Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(x) = ln(3 sin² 4x)
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.R.28
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.R.28Chapter 7, Problem 7.R.28
27–28. Curve sketching Use the graphing techniques of Section 4.4 to graph the following functions on their domains. Identify local extreme points, inflection points, concavity, and end behavior. Use a graphing utility only to check your work.
f(x) = ln x – ln² x
Verified step by step guidance1
Identify the domain of the function \(f(x) = \ln x - (\ln x)^2\). Since \(\ln x\) is defined only for \(x > 0\), the domain is \(x > 0\).
Find the first derivative \(f'(x)\) to determine critical points and intervals of increase or decrease. Use the chain rule and derivative of \(\ln x\):
\(f'(x) = \frac{1}{x} - 2 \ln x \cdot \frac{1}{x} = \frac{1 - 2 \ln x}{x}\).
Set the first derivative equal to zero to find critical points:
\(\frac{1 - 2 \ln x}{x} = 0 \implies 1 - 2 \ln x = 0 \implies \ln x = \frac{1}{2} \implies x = e^{\frac{1}{2}}\).
Also check where \(f'(x)\) is undefined, but since \(x > 0\), \(f'(x)\) is defined everywhere in the domain.
Find the second derivative \(f''(x)\) to analyze concavity and inflection points. Differentiate \(f'(x)\):
\(f''(x) = \frac{d}{dx} \left( \frac{1 - 2 \ln x}{x} \right)\).
Use the quotient or product rule carefully to express \(f''(x)\) in terms of \(x\) and \(\ln x\).
Set \(f''(x) = 0\) to find possible inflection points and analyze the sign of \(f''(x)\) on intervals to determine concavity:
- If \(f''(x) > 0\), the graph is concave up.
- If \(f''(x) < 0\), the graph is concave down.
Finally, analyze the end behavior by considering the limits as \(x \to 0^+\) and \(x \to \infty\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain and Logarithmic Functions
Understanding the domain of the function is crucial, especially since it involves natural logarithms (ln x). The function is defined only for x > 0 because the logarithm of non-positive numbers is undefined. Recognizing this domain restriction helps avoid errors in graphing and analysis.
Recommended video:
5:26
Graphs of Logarithmic Functions
Critical Points and Local Extrema
Finding local maxima and minima requires computing the first derivative and solving for points where it equals zero or is undefined. These critical points indicate potential local extreme values. Analyzing the sign changes of the derivative around these points confirms whether they are maxima, minima, or neither.
Recommended video:
04:50Critical Points
Concavity and Inflection Points
The second derivative reveals the concavity of the function—whether it curves upward or downward. Points where the second derivative changes sign are inflection points, indicating a change in concavity. Identifying these helps in sketching the curve accurately and understanding the function's behavior.
Recommended video:
05:59Determining Concavity Given a Function
Related Practice
Textbook Question
Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is
f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0
where ln x has zero mean and standard deviation σ > 0.
b. Evaluate lim x → 0 ƒ(x). (Hint: Let x = eʸ.)
Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(t) = cosh t sinh t
Textbook Question
37–56. Integrals Evaluate each integral.
∫ sech² w tanh w dw
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Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(x) = tanh⁻¹(cos x)
Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(x) = (sinh x) / (1 + sinh x)