Answer:
[tex]\displaystyle \frac{dy}{dx} = \frac{-4}{(2x - 1)^3}[/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 [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle y = \frac{1}{(2x - 1)^2}[/tex]
Step 2: Differentiate
- Rewrite: [tex]\displaystyle y = (2x - 1)^{-2}[/tex]
- Basic Power Rule [Derivative Rule - Chain Rule]: [tex]\displaystyle y' = -2(2x - 1)^{-3}(2x - 1)'[/tex]
- Basic Power Rule [Derivative Properties]: [tex]\displaystyle y' = -4(2x - 1)^{-3}[/tex]
- Rewrite: [tex]\displaystyle y' = \frac{-4}{(2x - 1)^3}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
