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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.6.33
Chapter 7, Problem 7.6.33

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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).
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Intro to the Chain Rule

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.
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Derivative of the Natural Logarithmic Function

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.
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Derivatives of Other Trig Functions
Related Practice
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