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

Verified step by step guidance

1. Identify the function given: \(y = \ln(\arctan(x))\). We need to find \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\).
2. Recall the chain rule for derivatives: if \(y = \ln(u)\), then \(\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}\). Here, \(u = \arctan(x)\).
3. Find the derivative of the inner function \(u = \arctan(x)\). The derivative is \(\frac{du}{dx} = \frac{1}{1 + x^2}\).
4. Apply the chain rule by substituting \(u\) and \(\frac{du}{dx}\) into the formula: \(\frac{dy}{dx} = \frac{1}{\arctan(x)} \cdot \frac{1}{1 + x^2}\).
5. Write the final expression for the derivative as \(\frac{dy}{dx} = \frac{1}{\arctan(x)(1 + x^2)}\).

Key Concepts

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.