Respuesta :

Space

Answer:

[tex]\displaystyle \frac{dy}{dx} = \frac{1}{x \Big( (\ln x)^2 + 1 \Big)}[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Rule [Chain Rule]:                                                                                 [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle y = \arctan (\ln x)[/tex]

Step 2: Differentiate

  1. Trigonometric Differentiation [Derivative Rule - Chain Rule]:                   [tex]\displaystyle y' = \frac{(\ln x)'}{(\ln x)^2 + 1}[/tex]
  2. Logarithmic Differentiation:                                                                         [tex]\displaystyle y' = \frac{\frac{1}{x}}{(\ln x)^2 + 1}[/tex]
  3. Simplify:                                                                                                         [tex]\displaystyle y' = \frac{1}{x \Big( (\ln x)^2 + 1 \Big)}[/tex]

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

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