In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
16. y = (ln x)³

검증된 단계별 안내

Identify the function given: \(y = (\ln x)^3\). This is a composite function where the outer function is \(u^3\) and the inner function is \(u = \ln x\).

Apply the chain rule for differentiation, which states that if \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\).

Differentiate the outer function \(u^3\) with respect to \(u\): \(\frac{d}{du}(u^3) = 3u^2\).

Differentiate the inner function \(u = \ln x\) with respect to \(x\): \(\frac{d}{dx}(\ln x) = \frac{1}{x}\).

Combine the results using the chain rule: \(\frac{dy}{dx} = 3(\ln x)^2 \cdot \frac{1}{x}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Chain Rule

The chain rule is used to differentiate composite functions. It states that the derivative of a function composed of another function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. For example, if y = (ln x)³, treat (ln x) as the inner function and cube as the outer function.

Derivative of the Natural Logarithm Function

The derivative of ln x with respect to x is 1/x. This is fundamental when differentiating expressions involving logarithms. Knowing this allows you to find the derivative of functions like (ln x)³ by applying the chain rule.

Power Rule

The power rule states that the derivative of xⁿ is n*xⁿ⁻¹. When differentiating (ln x)³, the power rule applies to the outer function (raising to the third power), which helps simplify the differentiation process when combined with the chain rule.

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.16. y = (ln x)³