Question

What is the derivative of x(sqrt (16-x^2)

Answer 1

`\displaystyle (dy)/(dx) = √(16 - x^2) - (x^2)/(√(16 - x^2))`

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 = x√(16 - x^2)`

Step 2: Differentiate

1. Derivative Rule [Product Rule]:
   `\displaystyle y' = (x)'√(16 - x^2) + x(√(16 - x^2))'`
2. Basic Power Rule [Derivative Rule - Chain Rule]:
   `\displaystyle y' = √(16 - x^2) + (x)/(2√(16 - x^2))(16 - x^2)'`
3. Basic Power Rule [Derivative Properties]:
   `\displaystyle y' = √(16 - x^2) - (x^2)/(2√(16 - x^2))`

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

Unit: Differentiation