Find y'' for the following functions.
y = cos θ sin θ

검증된 단계별 안내

First, recognize that the function y = cos(θ) sin(θ) is a product of two functions. To find the derivative, we will use the product rule, which states that if y = u*v, then y' = u'v + uv'.

Identify u = cos(θ) and v = sin(θ). Compute the derivatives u' and v'. The derivative of cos(θ) is -sin(θ), and the derivative of sin(θ) is cos(θ).

Apply the product rule: y' = (-sin(θ)) * sin(θ) + cos(θ) * cos(θ). Simplify this expression to get y'.

Now, to find y'', differentiate y' again. Use the sum rule and the chain rule as needed. Differentiate each term separately: for the first term, differentiate -sin(θ) * sin(θ), and for the second term, differentiate cos(θ) * cos(θ).

Combine the derivatives from the previous step to obtain y''. Simplify the expression to get the second derivative of y with respect to θ.

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주요 개념

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

Second Derivative

The second derivative of a function measures the rate of change of the first derivative, providing information about the curvature of the function's graph. It is denoted as y'' and is essential for analyzing the concavity and inflection points of the function. In this context, finding y'' involves differentiating the function twice with respect to the variable.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, especially when dealing with periodic phenomena. The function y = cos(θ) sin(θ) is a product of these functions, and understanding their derivatives requires applying the product rule. Familiarity with the properties and derivatives of sine and cosine is crucial for solving the problem.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states that if u and v are functions of θ, then the derivative of their product is given by u'v + uv'. This rule is particularly relevant for differentiating y = cos(θ) sin(θ), as it allows for the correct application of calculus to find the first and subsequently the second derivative.

Find y'' for the following functions.y = cos θ sin θ