Find y⁽⁴⁾ = d⁴y/dx⁴ if:

b. y = 9 cos x

검증된 단계별 안내

Start by identifying the function y = 9 cos(x). We need to find the fourth derivative of this function with respect to x.

Recall that the derivative of cos(x) is -sin(x). Therefore, the first derivative y' = dy/dx is y' = -9 sin(x).

Next, find the second derivative y'' = d²y/dx². The derivative of -sin(x) is -cos(x), so y'' = -9 cos(x).

Now, find the third derivative y''' = d³y/dx³. The derivative of -cos(x) is sin(x), so y''' = 9 sin(x).

Finally, find the fourth derivative y⁽⁴⁾ = d⁴y/dx⁴. The derivative of sin(x) is cos(x), so y⁽⁴⁾ = 9 cos(x).

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주요 개념

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

Higher-Order Derivatives

Higher-order derivatives refer to the derivatives of a function taken multiple times. The first derivative represents the rate of change of the function, the second derivative indicates the curvature or concavity, and so on. In this case, we are interested in the fourth derivative, which provides insights into the function's behavior beyond its initial slope and curvature.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe periodic phenomena. The function given, y = 9 cos x, is a cosine function scaled by a factor of 9. Understanding the properties of these functions, including their derivatives, is essential for solving problems involving them.

Derivative Rules

Derivative rules are formulas that simplify the process of finding derivatives of functions. For trigonometric functions, the derivative of cos x is -sin x. Applying these rules repeatedly allows us to compute higher-order derivatives efficiently, which is necessary for finding y⁽⁴⁾ in this problem.

Find y⁽⁴⁾ = d⁴y/dx⁴ if:b. y = 9 cos x