Question

Derivative of R=(100+50/lnx)

Answer 1

`\displaystyle R' = (-50)/(x(\ln x)^2)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]:
`\displaystyle (d)/(dx) [(f(x))/(g(x)) ]=(g(x)f'(x)-g'(x)f(x))/(g^2(x))`

Explanation:

Step 1: Define

Identify

`\displaystyle R = 100 + (50)/(\ln x)`

Step 2: Differentiate

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

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

Unit: Differentiation