Question

Find the derivative of (xlnx)/(1+lnx)

Answer 1

`\displaystyle (dy)/(dx) = 1 - (\ln x)/(\big( \ln (x) + 1 \big)^2)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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 [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

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 y = (x \ln x)/(1 + \ln x)`

Step 2: Differentiate

1. Derivative Rule [Quotient Rule]:
   `\displaystyle y' = ((x \ln x)'(1 + \ln x) - (x \ln x)(1 + \ln x)')/((1 + \ln x)^2)`
2. Derivative Rule [Product Rule]:
   `\displaystyle y' = ([(x)' \ln x + x(\ln x)'](1 + \ln x) - (x \ln x)(1 + \ln x)')/((1 + \ln x)^2)`
3. Logarithmic Differentiation [Derivative Properties]:
   `\displaystyle y' = ([(x)' \ln x + 1](1 + \ln x) - \ln x)/((1 + \ln x)^2)`
4. Basic Power Rule:
   `\displaystyle y' = (\big( \ln (x) + 1 \big) (1 + \ln x) - \ln x)/((1 + \ln x)^2)`
5. Simplify:
   `\displaystyle y' = ((1 + \ln x)^2 - \ln x)/((1 + \ln x)^2)`
6. Rewrite:
   `\displaystyle y' = ((1 + \ln x)^2)/((1 + \ln x)^2) - (\ln x)/((1 + \ln x)^2)`
7. Simplify:
   `\displaystyle y' = 1 - (\ln x)/((1 + \ln x)^2)`

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

Unit: Differentiation