Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(x))
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.PE.19
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
19. y = t arctan(t) - 1/2 ln(t)
Verified step by step guidance1
Identify the function to differentiate: \(y = t \arctan(t) - \frac{1}{2} \ln(t)\), where the variable is \(t\).
Apply the derivative operator \(\frac{dy}{dt}\) to each term separately: \(\frac{d}{dt} \left( t \arctan(t) \right)\) and \(\frac{d}{dt} \left( -\frac{1}{2} \ln(t) \right)\).
For the first term, use the product rule since it is a product of two functions of \(t\): \(u = t\) and \(v = \arctan(t)\). Recall the product rule: \(\frac{d}{dt}(uv) = u'v + uv'\).
Calculate the derivatives of the individual parts: \(u' = \frac{d}{dt}(t) = 1\) and \(v' = \frac{d}{dt}(\arctan(t)) = \frac{1}{1 + t^2}\).
For the second term, use the derivative of the natural logarithm: \(\frac{d}{dt} \left( -\frac{1}{2} \ln(t) \right) = -\frac{1}{2} \cdot \frac{1}{t}\). Combine all parts to write the full derivative expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Inverse Trigonometric Functions
The derivative of the arctan function, arctan(t), is 1/(1 + t²). Understanding how to differentiate inverse trigonometric functions is essential for handling terms like t arctan(t) in the expression.
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Derivatives of Other Inverse Trigonometric Functions
Product Rule
The product rule is used to differentiate products of two functions. For y = u(t)v(t), the derivative is u'v + uv'. This rule applies to the term t arctan(t), where both t and arctan(t) are functions of t.
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05:18The Product Rule
Derivative of Logarithmic Functions
The derivative of the natural logarithm ln(t) is 1/t. Recognizing this allows for straightforward differentiation of the term -1/2 ln(t), which is part of the given function.
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05:18Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–78.
35. ∫sec²x e^(tan x)dx
Textbook Question
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
f. f(x) = sech(x), g(x) = e^(-x)
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
23. y = arccsc(secθ), 0<θ<π/2
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
15. y = sin⁻¹√(1-u²), 0<u<1
Textbook Question
Evaluate the integrals in Exercises 31–78.
58. ∫(from 0 to ln9)e^θ(e^θ-1)^(1/2) dθ