Question

What is the derivative of y=ln(ln(x))?

Answer 1

`\displaystyle y' = (1)/(x \ln(x))`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

Explanation:

Step 1: Define

Identify

`\displaystyle y = \ln \Big( \ln(x) \Big)`

Step 2: Differentiate

1. Logarithmic Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle y' = (1)/(\ln(x)) \cdot (d)/(dx)[\ln(x)]`
2. Logarithmic Differentiation:
   `\displaystyle y' = (1)/(\ln(x)) \cdot (1)/(x)`
3. Simplify:
   `\displaystyle y' = (1)/(x \ln(x))`

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

Unit: Differentiation