Domanda del libro di testo
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))
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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 7.3.67
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"
Guida verificata passo dopo passo1
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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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.
Video consigliato:
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.
Video consigliato:
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.
Video consigliato:
06:35Derivatives of Other Inverse Trigonometric Functions
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