Question

Determine the second derivative of y=cosxsinx

Answer 1

`\displaystyle (dy)/(dx) = -4 \sin(x) \cos(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)`

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

Explanation:

Step 1: Define

Identify

`\displaystyle y = \cos(x) \sin(x)`

Step 2: Differentiate

1. Derivative Rule [Product Rule]:
   `\displaystyle y' = (d)/(dx)[\cos(x)] \sin(x) + \cos(x) (d)/(dx)[\sin(x)]`
2. Trigonometric Differentiation:
   `\displaystyle y' = -\sin^2(x) + \cos^2(x)`
3. Derivative Property [Addition/Subtraction]:
   `\displaystyle y' = (d)/(dx)[-\sin^2(x)] + (d)/(dx)[\cos^2(x)]`
4. Rewrite [Derivative Property - Multiplied Constant]:
   `\displaystyle y' = -(d)/(dx)[\sin^2(x)] + (d)/(dx)[\cos^2(x)]`
5. Basic Power Rule [Derivative Rule - Chain Rule]:
   `\displaystyle y' = -2 \sin(x) (d)/(dx)[\sin(x)] + 2 \cos(x) (d)/(dx)[\cos(x)]`
6. Trigonometric Differentiation:
   `\displaystyle y' = -2 \sin(x) \cos(x) - 2 \cos(x) \sin(x)`
7. Simplify:
   `\displaystyle y' = -4 \sin(x) \cos(x)`

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

Unit: Differentiation