In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
27. y = θ(sin(lnθ) + cos(lnθ))

검증된 단계별 안내

Identify the function given: \(y = \theta \left( \sin(\ln \theta) + \cos(\ln \theta) \right)\). We need to find \(\frac{dy}{d\theta}\) since the variable is \(\theta\).

Recognize that \(y\) is a product of two functions of \(\theta\): \(u = \theta\) and \(v = \sin(\ln \theta) + \cos(\ln \theta)\). We will use the product rule for differentiation: \(\frac{d}{d\theta}(uv) = u'v + uv'\).

Compute the derivative of \(u = \theta\), which is \(u' = 1\).

Compute the derivative of \(v = \sin(\ln \theta) + \cos(\ln \theta)\). Use the chain rule for each term: For \(\sin(\ln \theta)\), derivative is \(\cos(\ln \theta) \cdot \frac{1}{\theta}\); for \(\cos(\ln \theta)\), derivative is \(-\sin(\ln \theta) \cdot \frac{1}{\theta}\). Then sum these results to find \(v'\).

Apply the product rule: \(\frac{dy}{d\theta} = u'v + uv' = 1 \cdot \left( \sin(\ln \theta) + \cos(\ln \theta) \right) + \theta \cdot \left( \cos(\ln \theta) \cdot \frac{1}{\theta} - \sin(\ln \theta) \cdot \frac{1}{\theta} \right)\). Simplify the expression as much as possible.

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.

Chain Rule

The chain rule is used to differentiate composite functions. When a function is nested inside another, such as sin(lnθ), you differentiate the outer function first and multiply by the derivative of the inner function. This rule is essential for handling functions like sin(lnθ) and cos(lnθ).

Product Rule

The product rule is applied when differentiating the product of two functions, such as θ multiplied by (sin(lnθ) + cos(lnθ)). It states that the derivative of f(θ)g(θ) is f'(θ)g(θ) + f(θ)g'(θ), allowing you to differentiate each part separately and combine the results.

Derivatives of Logarithmic and Trigonometric Functions

Knowing the derivatives of lnθ, sin(x), and cos(x) is crucial. The derivative of lnθ is 1/θ, while the derivatives of sin(x) and cos(x) are cos(x) and -sin(x), respectively. These rules help differentiate the inner and outer functions in the expression.

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.27. y = θ(sin(lnθ) + cos(lnθ))