In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
31. y=arccot(√t)

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Identify the function to differentiate: \(y = \arccot(\sqrt{t})\). We need to find \(\frac{dy}{dt}\).

Recall the derivative formula for \(y = \arccot(u)\) with respect to \(u\): \(\frac{dy}{du} = -\frac{1}{1 + u^2}\).

Set \(u = \sqrt{t} = t^{1/2}\). Then find \(\frac{du}{dt}\) using the power rule: \(\frac{du}{dt} = \frac{1}{2} t^{-1/2}\).

Apply the chain rule: \(\frac{dy}{dt} = \frac{dy}{du} \cdot \frac{du}{dt} = -\frac{1}{1 + u^2} \cdot \frac{1}{2} t^{-1/2}\).

Substitute back \(u = \sqrt{t}\) into the expression to write the derivative entirely in terms of \(t\).

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주요 개념

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

Derivative of Inverse Trigonometric Functions

Inverse trigonometric functions like arccot(x) have specific derivative formulas. For arccot(x), the derivative with respect to x is -1/(1 + x²). Understanding these derivatives is essential to differentiate expressions involving inverse trig functions.

Chain Rule

The chain rule is used to differentiate composite functions. When y = arccot(√t), you first differentiate arccot(u) with respect to u, then multiply by the derivative of u = √t with respect to t. This rule allows handling nested functions effectively.

Derivative of Square Root Functions

The square root function √t can be rewritten as t^(1/2). Its derivative with respect to t is (1/2)t^(-1/2). Knowing how to differentiate root functions is necessary when applying the chain rule to expressions like arccot(√t).

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.31. y=arccot(√t)