In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
27. y=arccsc(x²+1)

검증된 단계별 안내

Identify the function to differentiate: \(y = \arccsc(x^{2} + 1)\), where the argument of the arccsc function is \(u = x^{2} + 1\).

Recall the derivative formula for \(y = \arccsc(u)\) with respect to \(x\): \(\frac{dy}{dx} = -\frac{1}{|u| \sqrt{u^{2} - 1}} \cdot \frac{du}{dx}\).

Compute the derivative of the inner function \(u = x^{2} + 1\): \(\frac{du}{dx} = 2x\).

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

Simplify the expression under the square root if possible, and write the final derivative expression in terms of \(x\).

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.

Derivative of Inverse Trigonometric Functions

Inverse trigonometric functions like arccsc(x) have specific derivative formulas. For arccsc(u), the derivative is -u' / (|u|√(u² - 1)), where u is a function of x. Understanding these formulas is essential to differentiate expressions involving inverse trig functions.

Chain Rule

The chain rule is used to differentiate composite functions. When y = f(g(x)), the derivative y' = f'(g(x)) * g'(x). In this problem, since the argument of arccsc is x² + 1, the chain rule helps differentiate the inner function x² + 1.

Simplification and Domain Considerations

After differentiation, simplifying the expression is important for clarity. Also, recognizing the domain restrictions of arccsc and its derivative, such as the argument being outside the interval (-1,1), ensures the derivative is valid and correctly interpreted.

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.27. y=arccsc(x²+1)