Textbook Question
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))
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 7.2.10
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
10. y = ln(t^(3/2))
Verified step by step guidance1
Recognize that the function is given as \(y = \ln\left(t^{\frac{3}{2}}\right)\), which is a natural logarithm of a power function.
Use the logarithmic property that allows you to bring the exponent in front: \(\ln\left(t^{\frac{3}{2}}\right) = \frac{3}{2} \ln(t)\).
Rewrite the function as \(y = \frac{3}{2} \ln(t)\) to simplify differentiation.
Recall the derivative of \(\ln(t)\) with respect to \(t\) is \(\frac{1}{t}\), so apply the constant multiple rule to get \(\frac{dy}{dt} = \frac{3}{2} \cdot \frac{1}{t}\).
Express the derivative as \(\frac{dy}{dt} = \frac{3}{2t}\), which is the derivative of the original function with respect to \(t\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Differentiation
Logarithmic differentiation involves applying the derivative rules to functions involving logarithms. The derivative of ln(u), where u is a function of the variable, is (1/u) times the derivative of u. This technique simplifies differentiation of functions expressed as logarithms.
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Logarithmic Differentiation
Power Rule for Exponents
The power rule states that the derivative of t^n with respect to t is n * t^(n-1). This rule is essential when differentiating expressions where the variable is raised to a constant exponent, such as t^(3/2) in the given function.
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Guided course
7:39Introduction to Exponent Rules
Properties of Logarithms
Logarithmic properties, like ln(a^b) = b * ln(a), allow simplification of expressions before differentiation. Applying these properties can transform complex logarithmic functions into simpler forms, making differentiation more straightforward.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 77–90.
79. ∫(from -1 to 0)6dt/√(3-2t-t²)
Textbook Question
Solve the initial value problems in Exercises 115–120.
117. dy/dx = 1/(x√(x² - 1)), x > 1; y(2) = π
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²
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Textbook Question
Evaluate the integrals in Exercises 87–96.
89. ∫₀¹ 2^(−θ) dθ
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