Textbook Question
Evaluate the integrals in Exercises 31–78.
49. ∫x3^(x²)dx
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 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
Chapter 7, Problem 7.PE.11
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
11. y = 5x^(3.6)
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Identify the function given: \(y = 5x^{3.6}\), where \(y\) is expressed in terms of \(x\).
Recall the power rule for differentiation: if \(y = ax^n\), then \(\frac{dy}{dx} = a n x^{n-1}\).
Apply the power rule to the function: multiply the coefficient 5 by the exponent 3.6 to get the new coefficient.
Subtract 1 from the exponent 3.6 to find the new exponent for \(x\) in the derivative.
Write the derivative as \(\frac{dy}{dx} = 5 \times 3.6 \times x^{3.6 - 1}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Rule for Differentiation
The power rule states that the derivative of x raised to a constant power n is n times x to the (n-1) power. It is expressed as d/dx[x^n] = n*x^(n-1), and applies to any real number exponent, including decimals.
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The Power Rule
Constant Multiple Rule
When differentiating a function multiplied by a constant, the constant can be factored out and multiplied by the derivative of the function. For example, d/dx[c*f(x)] = c * d/dx[f(x)], where c is a constant.
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04:02The Power Rule
Derivative Notation and Variables
Understanding the notation dy/dx means the derivative of y with respect to x, indicating the rate of change of y as x changes. Correctly identifying the variable of differentiation is essential for applying derivative rules properly.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–78.
33. ∫e^x sec²(e^x - 7)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
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
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Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
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Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
102. lim(x→0) (x sin(x²))/(tan³x)