Textbook Question
Evaluate the integrals in Exercises 97–110.
101. ∫ (log₁₀x / x) dx
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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 7.6.21
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
21. y=arccos(x²)
Verified step by step guidance1
Identify the function given: \(y = \arccos(x^{2})\). We want to find \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\).
Recall the derivative formula for the inverse cosine function: \(\frac{d}{dx}[\arccos(u)] = -\frac{1}{\sqrt{1 - u^{2}}} \cdot \frac{du}{dx}\), where \(u\) is a function of \(x\).
In this problem, set \(u = x^{2}\). Then compute \(\frac{du}{dx} = 2x\).
Substitute \(u = x^{2}\) and \(\frac{du}{dx} = 2x\) into the derivative formula to get: \(\frac{dy}{dx} = -\frac{1}{\sqrt{1 - (x^{2})^{2}}} \cdot 2x\).
Simplify the expression inside the square root: \((x^{2})^{2} = x^{4}\), so the derivative becomes \(\frac{dy}{dx} = -\frac{2x}{\sqrt{1 - x^{4}}}\).
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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
Inverse trigonometric functions like arccos(x) have specific derivative formulas. For arccos(x), the derivative is -1 divided by the square root of (1 - x²). Understanding this formula is essential to differentiate expressions involving arccos.
Recommended video:
06:35
Derivatives of Other Inverse Trigonometric Functions
Chain Rule
The chain rule is used to differentiate composite functions. When a function is composed of an outer function and an inner function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Recommended video:
05:02Intro to the Chain Rule
Power Rule
The power rule states that the derivative of xⁿ is n times x raised to the (n-1) power. This rule is necessary to differentiate the inner function x² in the given problem.
Recommended video:
Guided course
5:50Power Rules
Related Practice
Textbook Question
Suppose that the differentiable function y = f(x) has an inverse and that the graph of f passes through the point (2, 4) and has a slope of 1/3 there. Find the value of df⁻¹/dx at x = 4.
Textbook Question
11. Show that if positive functions f(x) and g(x) grow at the same rate as x→∞, then f=O(g) and g=O(f).
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Textbook Question
47. Carbon-14 The oldest known frozen human mummy, discovered in the Schnalstal glacier of the Italian Alps in 1991 and called Otzi, was found wearing straw shoes and a leather coat with goat fur, and holding a copper ax and stone dagger. It was estimated that Otzi died 5000 years before he was discovered in the melting glacier. How much of the original carbon-14 remained in Otzi at the time of his discovery?
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Textbook Question
Solve the differential equation in Exercises 9–22.
9. 2√(xy) (dy/dx) = 1, x, y > 0
Textbook Question
Evaluate the integrals in Exercises 53–76.
67. ∫dx/(2+(x-1)²)