Question

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

Answer 1

Answer:

$\displaystyle \frac{dy}{dx} = 1 - \frac{\ln x}{\big( \ln (x) + 1 \big)^2}$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹  

Derivative Rule [Product Rule]:                                                                             $\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

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

$\displaystyle y = \frac{x \ln x}{1 + \ln x}$

Step 2: Differentiate

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

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

Unit: Differentiation