Question

Please help with this AP Calculus question !

Answer 1

Answer:

C.  $\displaystyle \frac{cos(x)}{x} - ln(x)sin(x)$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Rule [Product Rule]:                                                                             $\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

Trig Derivatives

Logarithmic Derivatives

Step-by-step explanation:

Step 1: Define

Identify

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

Step 2: Differentiate

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

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

Unit: Differentiation

Book: College Calculus 10e