Question

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

Answer 1

Answer:

$\displaystyle y' = \frac{1}{x \ln(x)}$

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]:                                                                                 $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

Step 1: Define

Identify

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

Step 2: Differentiate

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

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

Unit: Differentiation