Answer:
[tex]\displaystyle R' = \frac{-50}{x(\ln x)^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 R = 100 + \frac{50}{\ln x}[/tex]
Step 2: Differentiate
- Derivative Property [Addition/Subtraction]: [tex]\displaystyle R' = \frac{d}{dx}[100] + \frac{d}{dx} \bigg[ \frac{50}{\ln x} \bigg][/tex]
- Rewrite [Derivative Property - Multiplied Constant]: [tex]\displaystyle R' = \frac{d}{dx}[100] + 50 \frac{d}{dx} \bigg[ \frac{1}{\ln x} \bigg][/tex]
- Basic Power Rule: [tex]\displaystyle R' = 50 \frac{d}{dx} \bigg[ \frac{1}{\ln x} \bigg][/tex]
- Derivative Rule [Quotient Rule]: [tex]\displaystyle R' = 50 \bigg(\frac{(1)' \ln x - (\ln x)'}{(\ln x)^2} \bigg)[/tex]
- Basic Power Rule: [tex]\displaystyle R' = 50 \bigg( \frac{-(\ln x)'}{(\ln x)^2} \bigg)[/tex]
- Logarithmic Differentiation: [tex]\displaystyle R' = 50 \bigg( \frac{\frac{-1}{x}}{(\ln x)^2} \bigg)[/tex]
- Simplify: [tex]\displaystyle R' = \frac{-50}{x(\ln x)^2}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
