Textbook Question
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
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 7.3.67
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"
Verified step by step guidance1
Identify the function to differentiate: \(y = 7^{\sec \theta} \ln 7\). Notice that \(\ln 7\) is a constant multiplier.
Rewrite the function to clarify the structure: \(y = (7^{\sec \theta}) \cdot (\ln 7)\), where \(\ln 7\) is constant with respect to \(\theta\).
Recall the derivative formula for an exponential function with a variable exponent: If \(y = a^{u(\theta)}\), then \(\frac{dy}{d\theta} = a^{u(\theta)} \ln a \cdot \frac{du}{d\theta}\).
Apply the formula with \(a = 7\) and \(u(\theta) = \sec \theta\). Compute \(\frac{du}{d\theta} = \frac{d}{d\theta} (\sec \theta) = \sec \theta \tan \theta\).
Combine all parts to write the derivative: \(\frac{dy}{d\theta} = \ln 7 \cdot 7^{\sec \theta} \cdot \ln 7 \cdot \sec \theta \tan \theta\). Simplify by multiplying constants where appropriate.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Exponential Functions with Variable Exponents
When differentiating functions where the exponent is a variable expression, such as a^g(x), we use the formula d/dx[a^g(x)] = a^g(x) * ln(a) * g'(x). This combines the chain rule with the natural logarithm of the base to handle the variable exponent.
Recommended video:
04:50
Derivatives of General Exponential Functions
Chain Rule
The chain rule is used to differentiate composite functions. It states that the derivative of f(g(x)) is f'(g(x)) * g'(x). In this problem, it applies to differentiating sec(θ) inside the exponent.
Recommended video:
05:02Intro to the Chain Rule
Derivative of Trigonometric Functions
Knowing the derivatives of trigonometric functions is essential. Specifically, the derivative of sec(θ) with respect to θ is sec(θ) tan(θ). This is needed to find g'(θ) when differentiating the exponent.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
63. y = x^π"
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 21–48, find the derivative of y with respect to the appropriate variable.
45. y=cos(x-arccos(x))
Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
23. y = ln(x)/(1+ln(x))
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Textbook Question
Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.
f(x) = 27x³