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.13

Chapter 7, Problem 7.R.13

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

f(t) = cosh t sinh t

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Recall the product rule for derivatives: if you have a function \(f(t) = u(t) v(t)\), then its derivative is \(f'(t) = u'(t) v(t) + u(t) v'(t)\).

Identify the two functions in the product: \(u(t) = \cosh t\) and \(v(t) = \sinh t\).

Find the derivatives of each function separately: \(\frac{d}{dt} \cosh t = \sinh t\) and \(\frac{d}{dt} \sinh t = \cosh t\).

Apply the product rule: \(f'(t) = (\sinh t)(\sinh t) + (\cosh t)(\cosh t)\).

Simplify the expression if possible, using hyperbolic identities such as \(\cosh^2 t - \sinh^2 t = 1\).

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Key Concepts

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

Hyperbolic Functions

Hyperbolic functions, such as sinh(t) and cosh(t), are analogs of trigonometric functions but based on hyperbolas. They have unique properties and identities, like cosh²(t) - sinh²(t) = 1, which are useful in differentiation and integration.

Product Rule for Differentiation

The product rule states that the derivative of a product of two functions u(t) and v(t) is u'(t)v(t) + u(t)v'(t). This rule is essential when differentiating functions like f(t) = cosh(t) * sinh(t), where both factors depend on t.

Derivatives of Hyperbolic Functions

The derivatives of basic hyperbolic functions are: d/dt[sinh(t)] = cosh(t) and d/dt[cosh(t)] = sinh(t). Knowing these derivatives allows you to apply the product rule correctly to find the derivative of functions involving hyperbolic terms.

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Asymptotes of Hyperbolas

Related Practice

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.

120

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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}

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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

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Textbook Question

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

f(x) = tanh⁻¹(cos x)

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. ln xy = (ln x)(ln y)

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Textbook Question

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