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))

검증된 단계별 안내

Identify the function given: \(y = \frac{\ln(x)}{1 + \ln(x)}\). We need to find \(\frac{dy}{dx}\) since the variable is \(x\).

Recognize that this is a quotient of two functions: numerator \(u = \ln(x)\) and denominator \(v = 1 + \ln(x)\).

Recall the quotient rule for derivatives: \(\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2}\).

Compute the derivatives of numerator and denominator separately: \(u' = \frac{d}{dx} \ln(x) = \frac{1}{x}\) and \(v' = \frac{d}{dx} (1 + \ln(x)) = \frac{1}{x}\).

Substitute \(u\), \(v\), \(u'\), and \(v'\) into the quotient rule formula to express \(\frac{dy}{dx}\) as \(\frac{(1 + \ln(x)) \cdot \frac{1}{x} - \ln(x) \cdot \frac{1}{x}}{(1 + \ln(x))^2}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Derivative of the Natural Logarithm Function

The derivative of ln(x) with respect to x is 1/x. This fundamental rule is essential when differentiating expressions involving natural logarithms, as it allows us to handle the logarithmic part of the function accurately.

Quotient Rule

The quotient rule is used to differentiate functions expressed as one function divided by another. It states that the derivative of f(x)/g(x) is (g(x)f'(x) - f(x)g'(x)) / [g(x)]², which is crucial for differentiating y = ln(x) / (1 + ln(x)).

Chain Rule

The chain rule helps differentiate composite functions by multiplying the derivative of the outer function by the derivative of the inner function. It is important here because ln(x) appears inside another function, requiring careful application of this rule.

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))