Question

Second derivative of y = x sinx

Answer 1

`\displaystyle (d^2y)/(dx^2) = 2 \cos (x) - x \sin (x)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

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'' = (d)/(dx)[\sin (x)] + (d)/(dx)[x \cos (x)]`
5. Derivative Rule [Product Rule]:
   `\displaystyle y'' = (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