교과서 질문
Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) dt
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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
7장, 문제 7.3.117
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
117. y = (√t)ᵗ
검증된 단계별 안내1
Rewrite the function to a form that is easier to differentiate using logarithmic differentiation. Start by expressing the function as \(y = (\sqrt{t})^{t} = (t^{1/2})^{t} = t^{t/2}\).
Take the natural logarithm of both sides to simplify the exponentiation: \(\ln y = \ln \left(t^{t/2}\right)\).
Use the logarithm power rule to bring down the exponent: \(\ln y = \frac{t}{2} \ln t\).
Differentiate both sides with respect to \(t\). Remember that \(y\) is a function of \(t\), so use implicit differentiation on the left side: \(\frac{1}{y} \frac{dy}{dt} = \frac{d}{dt} \left( \frac{t}{2} \ln t \right)\).
Apply the product rule to differentiate the right side: \(\frac{d}{dt} \left( \frac{t}{2} \ln t \right) = \frac{1}{2} \ln t + \frac{t}{2} \cdot \frac{1}{t}\). Then multiply both sides by \(y\) to solve for \(\frac{dy}{dt}\).
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions where the variable appears both in the base and the exponent. By taking the natural logarithm of both sides, the expression simplifies, allowing the use of implicit differentiation and product or chain rules more easily.
추천 영상:
06:30
Logarithmic Differentiation
Chain Rule
The chain rule is a fundamental differentiation rule used when differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
추천 영상:
05:02Intro to the Chain Rule
Properties of Logarithms
Properties of logarithms, such as log(a^b) = b log(a), help simplify expressions before differentiation. Applying these properties transforms complicated exponentials into products, making differentiation more straightforward.
추천 영상:
05:36Change of Base Property
관련 실천
교과서 질문
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
83. y = 3^(log₂ t)
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교과서 질문
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 3 - x, x < 0
= 3, x ≥ 0
32
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교과서 질문
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
39. lim (x → ∞) (ln 2x - ln(x + 1))
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교과서 질문
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
37. y=s√(1-s²) + arccos(s)
18
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교과서 질문
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
57. lim (x → 0⁺) x^(-1/ln x)
27
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