9–61. Evaluate and simplify y'.

y = 2x² cos^−1 x+ sin^−1 x

검증된 단계별 안내

First, identify the function y given in the problem. Here, y = 2x² cos⁻¹(x) + sin⁻¹(x).

To find y', the derivative of y with respect to x, apply the derivative rules to each term separately. Start with the term 2x² cos⁻¹(x).

For the term 2x² cos⁻¹(x), use the product rule: if u = 2x² and v = cos⁻¹(x), then y' = u'v + uv'. Calculate u' and v' separately.

The derivative of u = 2x² is u' = 4x. The derivative of v = cos⁻¹(x) is v' = -1/√(1-x²). Substitute these into the product rule formula.

Next, find the derivative of the second term sin⁻¹(x). The derivative of sin⁻¹(x) is 1/√(1-x²). Combine the derivatives of both terms to get y'.

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주요 개념

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

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the given function y to find y'.

Chain Rule

The Chain Rule is a technique used in differentiation when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This rule is essential for differentiating terms like cos^−1(x) and sin^−1(x) in the given expression.

Trigonometric Inverses

Trigonometric inverse functions, such as cos^−1(x) and sin^−1(x), are functions that return the angle whose cosine or sine is x, respectively. Understanding their derivatives is crucial for solving the problem, as they have specific derivative formulas that must be applied when differentiating the given function y.

9–61. Evaluate and simplify y'.y = 2x² cos^−1 x+ sin^−1 x