Answer:
C. [tex]\displaystyle \frac{cos(x)}{x} - ln(x)sin(x)[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Trig Derivatives
Logarithmic Derivatives
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle f(x) = ln(x)cos(x)[/tex]
Step 2: Differentiate
- Derivative Rule [Product Rule]: [tex]\displaystyle f'(x) = \frac{d}{dx}[ln(x)]cos(x) + ln(x)\frac{d}{dx}[cos(x)][/tex]
- Logarithmic Derivative: [tex]\displaystyle f'(x) = \frac{1}{x}cos(x) + ln(x)\frac{d}{dx}[cos(x)][/tex]
- Trig Derivative: [tex]\displaystyle f'(x) = \frac{1}{x}cos(x) + ln(x)[-sin(x)][/tex]
- Simplify: [tex]\displaystyle f'(x) = \frac{cos(x)}{x} - ln(x)sin(x)[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Book: College Calculus 10e
