Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.R.10

Chapter 7, Problem 7.R.10

10–19. Derivatives Find the derivatives of the following functions.
f(x) = ln(3 sin² 4x)

Verified step by step guidance

Recognize that the function is a composition of functions: \(f(x) = \ln(3 \sin^{2}(4x))\). We will need to use the chain rule and properties of logarithms to differentiate it.

Use the logarithm property to simplify the function before differentiating: \(\ln(3 \sin^{2}(4x)) = \ln(3) + \ln(\sin^{2}(4x)) = \ln(3) + 2 \ln(\sin(4x))\). Since \(\ln(3)\) is a constant, its derivative is zero.

Focus on differentiating \(2 \ln(\sin(4x))\). Apply the constant multiple rule: the derivative is \(2\) times the derivative of \(\ln(\sin(4x))\).

Differentiate \(\ln(\sin(4x))\) using the chain rule: the derivative of \(\ln(u)\) is \(\frac{1}{u} \cdot \frac{du}{dx}\). Here, \(u = \sin(4x)\), so \(\frac{du}{dx} = \cos(4x) \cdot 4\) by the chain rule.

Combine all parts to write the derivative: \(f'(x) = 2 \cdot \frac{1}{\sin(4x)} \cdot \cos(4x) \cdot 4\). Simplify this expression as much as possible to get the final derivative.

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Key Concepts

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

Chain Rule

The chain rule is used to differentiate composite functions by taking the derivative of the outer function evaluated at the inner function and multiplying it by the derivative of the inner function. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). This rule is essential when differentiating functions like ln(3 sin² 4x), which involve nested functions.

Derivative of the Natural Logarithm Function

The derivative of ln(u), where u is a differentiable function of x, is (1/u) * du/dx. This means you first find the derivative of the inside function u and then divide by u itself. This property simplifies differentiating logarithmic functions such as ln(3 sin² 4x).

Derivative of Trigonometric Functions and Power Rule

To differentiate expressions like sin²(4x), you apply the power rule and the chain rule together. The power rule states that d/dx [u^n] = n u^(n-1) du/dx, and the derivative of sin(4x) involves the chain rule: cos(4x) * 4. Combining these helps find the derivative of sin²(4x).

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Related Practice

Textbook Question

63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.

∫₁/₈¹ dx/x√(1 + x²/³)

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

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

Textbook Question

37–56. Integrals Evaluate each integral.

∫ sech² w tanh w dw

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Textbook Question

37–56. Integrals Evaluate each integral.

∫ tanh²x dx (Hint: Use an identity.)

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Textbook Question

10–19. Derivatives Find the derivatives of the following functions.

f(x) = (sinh x) / (1 + sinh x)