Question

second derivative of y = x sinx

Answer 1

Answer:

$\displaystyle \frac{d^2y}{dx^2} = 2 \cos (x) - x \sin (x)$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle y = x \sin (x)$

Step 2: Differentiate

1. Derivative Rule [Product Rule]:                                                                     $\displaystyle y' = (x)' \sin (x) + x \big( \sin (x) \big)'$
2. Basic Power Rule:                                                                                         $\displaystyle y' = \sin (x) + x \big( \sin (x) \big)'$
3. Trigonometric Differentiation:                                                                      $\displaystyle y' = \sin (x) + x \cos (x)$
4. Derivative Property [Addition/Subtraction]:                                                 $\displaystyle y'' = \frac{d}{dx}[\sin (x)] + \frac{d}{dx}[x \cos (x)]$
5. Derivative Rule [Product Rule]:                                                                     $\displaystyle y'' = \frac{d}{dx}[\sin (x)] + (x)' \cos (x) + x \big( \cos (x) \big)'$
6. Trigonometric Differentiation:                                                                       $\displaystyle y'' = \cos (x) + (x)' \cos (x) - x \sin (x)$
7. Basic Power Rule:                                                                                         $\displaystyle y'' = \cos (x) + \cos (x) - x \sin (x)$
8. Simplify:                                                                                                         $\displaystyle y'' = 2\cos (x) - x \sin (x)$

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

Unit: Differentiation