Answer:
\(\displaystyle \frac{d}{dx} = \frac{\sqrt{6x}}{x}\)
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: \(\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)\)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Chain Rule]: \(\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)\)
Step-by-step explanation:
Step 1: Define
Identify
\(\displaystyle y = \sqrt{24x}\)
Step 2: Differentiate
- Basic Power Rule [Derivative Rule - Chain Rule]: \(\displaystyle y' = \frac{1}{2\sqrt{24x}} \cdot \frac{d}{dx}[24x]\)
- Basic Power Rule [Derivative Property - Multiplied Constant]: \(\displaystyle y' = \frac{24}{2\sqrt{24x}}\)
- Simplify: \(\displaystyle y' = \frac{12}{\sqrt{24x}}\)
- Rationalize: \(\displaystyle y' = \frac{12\sqrt{24x}}{24x}\)
- Simplify: \(\displaystyle y' = \frac{\sqrt{6x}}{x}\)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
