Question

Find derivative of the average profit function.

Answer 1

`\displaystyle P'(x) = (60)/((4x + 5)^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 [Quotient Rule]:
`\displaystyle (d)/(dx) [(f(x))/(g(x)) ]=(g(x)f'(x)-g'(x)f(x))/(g^2(x))`

Explanation:

Step 1: Define

Identify

`\displaystyle P(x) = (8x - 5)/(4x + 5)`

Step 2: Differentiate

1. Derivative Rule [Quotient Rule]:
   `\displaystyle P'(x) = ((8x - 5)'(4x + 5) - (8x - 5)(4x + 5)')/((4x + 5)^2)`
2. Basic Power Rule [Derivative Properties]:
   `\displaystyle P'(x) = (8(4x + 5) - 4(8x - 5))/((4x + 5)^2)`
3. Expand:
   `\displaystyle P'(x) = (32x + 40 - 32x + 20)/((4x + 5)^2)`
4. Simplify:
   `\displaystyle P'(x) = (60)/((4x + 5)^2)`

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

Unit: Differentiation