Question

Please help with this AP Calculus question !

Answer 1

C.
`\displaystyle (cos(x))/(x) - ln(x)sin(x)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

Trig Derivatives

Logarithmic Derivatives

Explanation:

Step 1: Define

Identify

`\displaystyle f(x) = ln(x)cos(x)`

Step 2: Differentiate

1. Derivative Rule [Product Rule]:
   `\displaystyle f'(x) = (d)/(dx)[ln(x)]cos(x) + ln(x)(d)/(dx)[cos(x)]`
2. Logarithmic Derivative:
   `\displaystyle f'(x) = (1)/(x)cos(x) + ln(x)(d)/(dx)[cos(x)]`
3. Trig Derivative:
   `\displaystyle f'(x) = (1)/(x)cos(x) + ln(x)[-sin(x)]`
4. Simplify:
   `\displaystyle f'(x) = (cos(x))/(x) - ln(x)sin(x)`

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

Unit: Differentiation

Book: College Calculus 10e