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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.R.18
Chapter 7, Problem 7.R.18

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


g(t) = sinh⁻¹(√t)

Verified step by step guidance
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Recall that the function given is \( g(t) = \sinh^{-1}(\sqrt{t}) \), where \( \sinh^{-1}(x) \) is the inverse hyperbolic sine function, also written as \( \text{arsinh}(x) \).
Use the chain rule to differentiate \( g(t) \). The chain rule states that if \( g(t) = f(h(t)) \), then \( g'(t) = f'(h(t)) \cdot h'(t) \). Here, \( f(x) = \sinh^{-1}(x) \) and \( h(t) = \sqrt{t} = t^{1/2} \).
Find the derivative of the outer function \( f(x) = \sinh^{-1}(x) \). The derivative is \( f'(x) = \frac{1}{\sqrt{x^2 + 1}} \).
Find the derivative of the inner function \( h(t) = t^{1/2} \). Using the power rule, \( h'(t) = \frac{1}{2} t^{-1/2} = \frac{1}{2\sqrt{t}} \).
Combine the results using the chain rule: \( g'(t) = f'(h(t)) \cdot h'(t) = \frac{1}{\sqrt{(\sqrt{t})^2 + 1}} \cdot \frac{1}{2\sqrt{t}} \). Simplify the expression inside the square root and write the final derivative expression.

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Key Concepts

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

Inverse Hyperbolic Sine Function (sinh⁻¹)

The inverse hyperbolic sine function, sinh⁻¹(x), returns the value whose hyperbolic sine is x. It can be expressed as ln(x + √(x² + 1)), which is useful for differentiation. Understanding its definition helps in applying derivative rules correctly.
Recommended video:
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Inverse Sine

Chain Rule

The chain rule is a fundamental differentiation technique used when a function is composed of other functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This is essential for differentiating g(t) = sinh⁻¹(√t).
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Intro to the Chain Rule

Derivative of Square Root Function

The square root function, √t, can be rewritten as t^(1/2). Its derivative is (1/2)t^(-1/2), which is necessary when applying the chain rule to functions involving square roots. Recognizing this derivative simplifies the differentiation process.
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Root Test
Related Practice
Textbook Question

Moore’s Law In 1965, Gordon Moore observed that the number of transistors that could be placed on an integrated circuit was approximately doubling each year, and he predicted that this trend would continue for another decade. In 1975, Moore revised the doubling time to every two years, and this prediction became known as Moore’s Law.


a. In 1979, Intel introduced the Intel 8088 processor; each of its integrated circuits contained 29,000 transistors. Use Moore’s revised doubling time to find a function y(t) that approximates the number of transistors on an integrated circuit t years after 1979.

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f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0

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e. For what value of σ > 0 in part (d) does ƒ(x*) have a minimum?

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b. d/dx (x sech x)

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