Answer:
\(\displaystyle R' = \frac{-50}{x(\ln x)^2}\)
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: \(\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)\)
Derivative Property [Addition/Subtraction]: \(\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]\)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Quotient Rule]: \(\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}\)
Step-by-step explanation:
Step 1: Define
Identify
\(\displaystyle R = 100 + \frac{50}{\ln x}\)
Step 2: Differentiate
- Derivative Property [Addition/Subtraction]: \(\displaystyle R' = \frac{d}{dx}[100] + \frac{d}{dx} \bigg[ \frac{50}{\ln x} \bigg]\)
- Rewrite [Derivative Property - Multiplied Constant]: \(\displaystyle R' = \frac{d}{dx}[100] + 50 \frac{d}{dx} \bigg[ \frac{1}{\ln x} \bigg]\)
- Basic Power Rule: \(\displaystyle R' = 50 \frac{d}{dx} \bigg[ \frac{1}{\ln x} \bigg]\)
- Derivative Rule [Quotient Rule]: \(\displaystyle R' = 50 \bigg(\frac{(1)' \ln x - (\ln x)'}{(\ln x)^2} \bigg)\)
- Basic Power Rule: \(\displaystyle R' = 50 \bigg( \frac{-(\ln x)'}{(\ln x)^2} \bigg)\)
- Logarithmic Differentiation: \(\displaystyle R' = 50 \bigg( \frac{\frac{-1}{x}}{(\ln x)^2} \bigg)\)
- Simplify: \(\displaystyle R' = \frac{-50}{x(\ln x)^2}\)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
