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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.35
Chapter 7, Problem 7.3.35

22–36. Derivatives Find the derivatives of the following functions.


f(x) = x sinh⁻¹ x − √(x² + 1)

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Step 1: Recognize that the function f(x) = x sinh⁻¹(x) − √(x² + 1) involves the inverse hyperbolic sine function (sinh⁻¹(x)) and a square root term. To find the derivative, we will apply the product rule, chain rule, and the derivative of inverse hyperbolic functions.
Step 2: Recall the derivative of sinh⁻¹(x), which is d/dx [sinh⁻¹(x)] = 1 / √(x² + 1). Also, recall the derivative of √(x² + 1), which is d/dx [√(x² + 1)] = (1 / (2√(x² + 1))) * d/dx [x² + 1] = x / √(x² + 1).
Step 3: Apply the product rule to the term x sinh⁻¹(x). The product rule states that d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x). Here, u(x) = x and v(x) = sinh⁻¹(x). Compute u'(x) = 1 and v'(x) = 1 / √(x² + 1). Substitute these into the product rule.
Step 4: Differentiate the second term, −√(x² + 1), using the chain rule. As derived earlier, the derivative of √(x² + 1) is x / √(x² + 1). Therefore, the derivative of −√(x² + 1) is −x / √(x² + 1).
Step 5: Combine all the results. The derivative of f(x) is f'(x) = [1 * sinh⁻¹(x) + x * (1 / √(x² + 1))] − (x / √(x² + 1)). Simplify the expression 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.

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about 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 chain rule, depending on the form of the function.
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Derivatives

Inverse Hyperbolic Functions

The inverse hyperbolic sine function, denoted as sinh⁻¹(x), is the inverse of the hyperbolic sine function. It is defined for all real numbers and is useful in calculus for finding derivatives and integrals involving hyperbolic functions. Understanding how to differentiate inverse hyperbolic functions is essential for solving problems that involve them.
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Square Root Function

The square root function, represented as √(x² + 1), is a common mathematical function that returns the non-negative square root of its argument. When differentiating this function, the chain rule is often applied, as it involves a composition of functions. Recognizing how to handle square roots in differentiation is crucial for accurately finding derivatives of more complex expressions.
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Related Practice
Textbook Question

37–56. Integrals Evaluate each integral.

∫ dx/x√(16 + x²)

Textbook Question

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.


∫ (x²) / (4x³ + 7) dx

Textbook Question

Surface area of a catenoid When the catenary y = a cosh x/a is revolved about the x-axis, it sweeps out a surface of revolution called a catenoid. Find the area of the surface generated when y = cosh x on [–ln 2, ln 2] is rotated about the x-axis.

Textbook Question

88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.


lim x → ∞ (1 − coth x) / (1 − tanh x)

Textbook Question

63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.


limₕ→₀ (1 + 3h)^{2/h}

Textbook Question

Tsunamis A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging from 150 to 1000 km. Imagine a tsunami traveling across the Pacific Ocean, which is the deepest ocean in the world, with an average depth of about 4000 m. Explain why the shallow-water velocity equation (Exercise 75) applies to tsunamis even though the actual depth of the water is large. What does the shallow-water equation say about the speed of a tsunami in the Pacific Ocean (use d = 4000 m)?