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derivative of R=(100+50/lnx)

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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:

  1. f(x) = cxⁿ
  2. 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

  1. Derivative Property [Addition/Subtraction]:                                                 [tex]\displaystyle R' = \frac{d}{dx}[100] + \frac{d}{dx} \bigg[ \frac{50}{\ln x} \bigg][/tex]
  2. Rewrite [Derivative Property - Multiplied Constant]:                                   [tex]\displaystyle R' = \frac{d}{dx}[100] + 50 \frac{d}{dx} \bigg[ \frac{1}{\ln x} \bigg][/tex]
  3. Basic Power Rule:                                                                                         [tex]\displaystyle R' = 50 \frac{d}{dx} \bigg[ \frac{1}{\ln x} \bigg][/tex]
  4. Derivative Rule [Quotient Rule]:                                                                   [tex]\displaystyle R' = 50 \bigg(\frac{(1)' \ln x - (\ln x)'}{(\ln x)^2} \bigg)[/tex]
  5. Basic Power Rule:                                                                                         [tex]\displaystyle R' = 50 \bigg( \frac{-(\ln x)'}{(\ln x)^2} \bigg)[/tex]
  6. Logarithmic Differentiation:                                                                         [tex]\displaystyle R' = 50 \bigg( \frac{\frac{-1}{x}}{(\ln x)^2} \bigg)[/tex]
  7. Simplify:                                                                                                         [tex]\displaystyle R' = \frac{-50}{x(\ln x)^2}[/tex]

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

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