Answer:
[tex]\displaystyle \frac{dy}{dx} = \frac{3}{x \ln x^3}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
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 = \ln (\ln x^3)[/tex]
Step 2: Differentiate
- Logarithmic Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle y' = \frac{1}{\ln x^3} \cdot \frac{d}{dx}[\ln x^3][/tex]
- Logarithmic Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle y' = \frac{1}{\ln x^3} \cdot \frac{1}{x^3} \cdot \frac{d}{dx}[x^3][/tex]
- Basic Power Rule: [tex]\displaystyle y' = \frac{1}{\ln x^3} \cdot \frac{1}{x^3} \cdot 3x^2[/tex]
- Simplify: [tex]\displaystyle y' = \frac{3}{x \ln x^3}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
