Respuesta :

Space

Answer:

[tex]\displaystyle \frac{dy}{dx} = 2 \tan (x)[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                          [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

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 = 2 \ln (\sec x)[/tex]

Step 2: Differentiate

  1. Logarithmic Differentiation [Chain Rule, Multiplied Constant]:                   [tex]\displaystyle \frac{dy}{dx} = 2 \bigg( \frac{1}{\sec x} \bigg) \cdot \frac{d}{dx}[\sec x][/tex]
  2. Trigonometric Differentiation:                                                                       [tex]\displaystyle \frac{dy}{dx} = 2 \bigg( \frac{1}{\sec x} \bigg) \cdot \sec x \tan x[/tex]
  3. Simplify:                                                                                                         [tex]\displaystyle \frac{dy}{dx} = 2 \tan (x)[/tex]

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

Unit: Differentiation

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