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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 7 - Transcendental Functions문제 7.3.120
7장, 문제 7.3.120

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.
120. y = x^(sin x)

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Recognize that the function is of the form \(y = x^{\sin x}\), where both the base and the exponent depend on \(x\). This suggests using logarithmic differentiation to simplify the differentiation process.
Take the natural logarithm of both sides to get \(\ln y = \ln \left( x^{\sin x} \right)\). Using the logarithm power rule, rewrite this as \(\ln y = \sin x \cdot \ln x\).
Differentiate both sides with respect to \(x\). On the left side, use implicit differentiation: \(\frac{1}{y} \frac{dy}{dx}\). On the right side, apply the product rule to \(\sin x \cdot \ln x\).
Apply the product rule: \(\frac{d}{dx} (\sin x \cdot \ln x) = \cos x \cdot \ln x + \sin x \cdot \frac{1}{x}\).
Solve for \(\frac{dy}{dx}\) by multiplying both sides by \(y\), and then substitute back \(y = x^{\sin x}\) to express the derivative in terms of \(x\).

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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, such as y = x^(sin x). By taking the natural logarithm of both sides, the expression simplifies, allowing the use of implicit differentiation to find the derivative more easily.
추천 영상:
06:30
Logarithmic Differentiation

Chain Rule

The chain rule is a fundamental differentiation rule used when differentiating composite functions. In this problem, it is essential for differentiating expressions like sin(x) and the logarithm of x, as these functions are nested within each other after applying logarithmic differentiation.
추천 영상:
05:02
Intro to the Chain Rule

Implicit Differentiation

Implicit differentiation involves differentiating both sides of an equation with respect to the independent variable when y is defined implicitly. After taking the logarithm of y, implicit differentiation helps find dy/dx by treating y as a function of x and differentiating accordingly.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative
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