Textbook Question
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x² − 2x, x ≤ 1
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 7.3.63
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
63. y = x^π"
Verified step by step guidance1
Identify the function given: \(y = x^{\pi}\), where \(\pi\) is a constant approximately equal to 3.14159.
Recall the power rule for differentiation when the exponent is a constant: if \(y = x^n\), then \(\frac{dy}{dx} = n x^{n-1}\).
Apply the power rule to the function: since \(n = \pi\), the derivative is \(\frac{dy}{dx} = \pi x^{\pi - 1}\).
Write the final expression for the derivative without simplifying the exponent further: \(\frac{dy}{dx} = \pi x^{\pi - 1}\).
Note that this derivative is valid for \(x > 0\) because the base \(x\) is raised to an irrational power \(\pi\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Power Functions
The derivative of a power function y = x^n, where n is a constant, is found using the power rule: dy/dx = n * x^(n-1). This rule applies when the exponent is any real number, including irrational constants like π.
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07:32
Representing Functions as Power Series
Constant Exponents and Irrational Numbers
When the exponent is a constant, even if irrational like π, it is treated as a fixed number during differentiation. This means the power rule applies directly without additional steps, simplifying the process.
Recommended video:
Guided course
7:39Introduction to Exponent Rules
Notation and Differentiation with Respect to the Independent Variable
Differentiating y with respect to x means finding dy/dx, the rate at which y changes as x changes. Understanding this notation is essential for applying differentiation rules correctly to functions of x.
Recommended video:
05:53Finding Differentials
Related Practice
Textbook Question
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=x²+1, x≥0
Textbook Question
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
43. y=√(arcsin x)
Textbook Question
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
45. y=cos(x-arccos(x))