Domanda del libro di testo
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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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.6.21
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
21. y=arccos(x²)
Guida verificata passo dopo passo1
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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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.
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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.
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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.
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5:50Power Rules
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