Question

Derivative of 2ln(secx)

Answer 1

`\displaystyle (dy)/(dx) = 2 \tan (x)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle y = 2 \ln (\sec x)`

Step 2: Differentiate

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

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

Unit: Differentiation