In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
33. y=ln(arctan(x))

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Identify the function given: \(y = \ln(\arctan(x))\). We need to find \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\).

Recall the chain rule for derivatives: if \(y = \ln(u)\), then \(\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}\). Here, \(u = \arctan(x)\).

Find the derivative of the inner function \(u = \arctan(x)\). The derivative is \(\frac{du}{dx} = \frac{1}{1 + x^2}\).

Apply the chain rule by substituting \(u\) and \(\frac{du}{dx}\) into the formula: \(\frac{dy}{dx} = \frac{1}{\arctan(x)} \cdot \frac{1}{1 + x^2}\).

Write the final expression for the derivative as \(\frac{dy}{dx} = \frac{1}{\arctan(x)(1 + x^2)}\).

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주요 개념

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

Chain Rule

The chain rule is a fundamental differentiation technique used when a function is composed of two or more functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Derivative of the Natural Logarithm Function

The derivative of the natural logarithm function ln(u), where u is a differentiable function of x, is given by (1/u) * du/dx. This means you first find the derivative of the inside function u, then divide by u itself. This rule is essential when differentiating logarithmic expressions involving composite functions.

Derivative of the Arctangent Function

The derivative of arctan(x) with respect to x is 1/(1 + x^2). This formula is crucial when differentiating expressions involving inverse trigonometric functions. When arctan(x) appears inside another function, its derivative must be combined with the chain rule.

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.33. y=ln(arctan(x))