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.17

Chapter 7, Problem 7.R.17

10–19. Derivatives Find the derivatives of the following functions.
f(x) = tanh⁻¹(cos x)

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Recognize that the function is a composition of two functions: the inverse hyperbolic tangent function, \(\tanh^{-1}(u)\), where \(u = \cos x\). We will need to use the chain rule to differentiate it.

Recall the derivative of the inverse hyperbolic tangent function: \(\frac{d}{du} \tanh^{-1}(u) = \frac{1}{1 - u^2}\), valid for \(|u| < 1\).

Apply the chain rule: \(\frac{d}{dx} \tanh^{-1}(\cos x) = \frac{1}{1 - (\cos x)^2} \cdot \frac{d}{dx}(\cos x)\).

Find the derivative of the inner function: \(\frac{d}{dx}(\cos x) = -\sin x\).

Combine the results to write the derivative as \(f'(x) = \frac{-\sin x}{1 - \cos^2 x}\). You can simplify the denominator using the Pythagorean identity if desired.

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Key Concepts

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

Inverse Hyperbolic Tangent Function (tanh⁻¹)

The inverse hyperbolic tangent function, tanh⁻¹(x), returns the value whose hyperbolic tangent is x. It is defined for |x| < 1 and has the derivative d/dx [tanh⁻¹(x)] = 1 / (1 - x²). Understanding its domain and derivative formula is essential for differentiating compositions involving tanh⁻¹.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). Applying the chain rule correctly allows us to differentiate tanh⁻¹(cos x) by combining the derivative of tanh⁻¹ with that of cos x.

Derivative of Cosine Function

The cosine function, cos x, is a basic trigonometric function whose derivative is -sin x. Knowing this derivative is crucial when applying the chain rule to functions like tanh⁻¹(cos x), as it provides the inner function's rate of change needed for the overall derivative.

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Derivatives of Sine & Cosine

Related Practice

Textbook Question

Limit Evaluate lim x → ∞ (tanh x)ˣ.

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

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

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)

Textbook Question

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

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

10–19. Derivatives Find the derivatives of the following functions.f(x) = tanh⁻¹(cos x)

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Key Concepts

Inverse Hyperbolic Tangent Function (tanh⁻¹)

Chain Rule

Derivative of Cosine Function

10–19. Derivatives Find the derivatives of the following functions.
f(x) = tanh⁻¹(cos x)