In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
21. y=arccos(x²)

검증된 단계별 안내

Identify the function given: \(y = \arccos(x^{2})\). We want to find \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\).

Recall the derivative formula for the inverse cosine function: \(\frac{d}{dx}[\arccos(u)] = -\frac{1}{\sqrt{1 - u^{2}}} \cdot \frac{du}{dx}\), where \(u\) is a function of \(x\).

In this problem, set \(u = x^{2}\). Then compute \(\frac{du}{dx} = 2x\).

Substitute \(u = x^{2}\) and \(\frac{du}{dx} = 2x\) into the derivative formula to get: \(\frac{dy}{dx} = -\frac{1}{\sqrt{1 - (x^{2})^{2}}} \cdot 2x\).

Simplify the expression inside the square root: \((x^{2})^{2} = x^{4}\), so the derivative becomes \(\frac{dy}{dx} = -\frac{2x}{\sqrt{1 - x^{4}}}\).

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.

Derivative of Inverse Trigonometric Functions

Inverse trigonometric functions like arccos(x) have specific derivative formulas. For arccos(x), the derivative is -1 divided by the square root of (1 - x²). Understanding this formula is essential to differentiate expressions involving arccos.

Chain Rule

The chain rule is used to differentiate composite functions. When a function is composed of an outer function and an inner function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Power Rule

The power rule states that the derivative of xⁿ is n times x raised to the (n-1) power. This rule is necessary to differentiate the inner function x² in the given problem.

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.21. y=arccos(x²)