Answer:
[tex]\displaystyle P'(x) = \frac{60}{(4x + 5)^2}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Quotient Rule]: [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle P(x) = \frac{8x - 5}{4x + 5}[/tex]
Step 2: Differentiate
- Derivative Rule [Quotient Rule]: [tex]\displaystyle P'(x) = \frac{(8x - 5)'(4x + 5) - (8x - 5)(4x + 5)'}{(4x + 5)^2}[/tex]
- Basic Power Rule [Derivative Properties]: [tex]\displaystyle P'(x) = \frac{8(4x + 5) - 4(8x - 5)}{(4x + 5)^2}[/tex]
- Expand: [tex]\displaystyle P'(x) = \frac{32x + 40 - 32x + 20}{(4x + 5)^2}[/tex]
- Simplify: [tex]\displaystyle P'(x) = \frac{60}{(4x + 5)^2}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
