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Identical expressions
(lnx^ two - two x)/(lnx^2-2lnx)
(lnx squared minus 2x) divide by (lnx squared minus 2lnx)
(lnx to the power of two minus two x) divide by (lnx squared minus 2lnx)
(lnx2-2x)/(lnx2-2lnx)
lnx2-2x/lnx2-2lnx
(lnx²-2x)/(lnx²-2lnx)
(lnx to the power of 2-2x)/(lnx to the power of 2-2lnx)
lnx^2-2x/lnx^2-2lnx
(lnx^2-2x) divide by (lnx^2-2lnx)
Similar expressions
(lnx^2-2x)/(lnx^2+2lnx)
(lnx^2+2x)/(lnx^2-2lnx)

The derivative of the function/ (lnx^2-2x)/(lnx^2-2lnx)

Derivative of (lnx^2-2x)/(lnx^2-2lnx)

The solution

You have entered [src]

     2            
  log (x) - 2*x   
------------------
   2              
log (x) - 2*log(x)

$$\frac{- 2 x + \log{\left(x \right)}^{2}}{\log{\left(x \right)}^{2} - 2 \log{\left(x \right)}}$$

(log(x)^2 - 2*x)/(log(x)^2 - 2*log(x))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = - 2 x + \log{\left(x \right)}^{2}$ and $g{\left(x \right)} = \log{\left(x \right)}^{2} - 2 \log{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $- 2 x + \log{\left(x \right)}^{2}$ term by term:
  1. Let $u = \log{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result of the chain rule is:
    
    $\frac{2 \log{\left(x \right)}}{x}$
  4. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $-2$
  The result is: $-2 + \frac{2 \log{\left(x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\log{\left(x \right)}^{2} - 2 \log{\left(x \right)}$ term by term:
  1. Let $u = \log{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result of the chain rule is:
    
    $\frac{2 \log{\left(x \right)}}{x}$
  4. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    So, the result is: $- \frac{2}{x}$
  The result is: $\frac{2 \log{\left(x \right)}}{x} - \frac{2}{x}$
Now plug in to the quotient rule:

$\frac{\left(-2 + \frac{2 \log{\left(x \right)}}{x}\right) \left(\log{\left(x \right)}^{2} - 2 \log{\left(x \right)}\right) - \left(- 2 x + \log{\left(x \right)}^{2}\right) \left(\frac{2 \log{\left(x \right)}}{x} - \frac{2}{x}\right)}{\left(\log{\left(x \right)}^{2} - 2 \log{\left(x \right)}\right)^{2}}$
Now simplify:

$\frac{2 \left(- \left(x - \log{\left(x \right)}\right) \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + \left(2 x - \log{\left(x \right)}^{2}\right) \left(\log{\left(x \right)} - 1\right)\right)}{x \left(\log{\left(x \right)} - 2\right)^{2} \log{\left(x \right)}^{2}}$

The answer is:

$\frac{2 \left(- \left(x - \log{\left(x \right)}\right) \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + \left(2 x - \log{\left(x \right)}^{2}\right) \left(\log{\left(x \right)} - 1\right)\right)}{x \left(\log{\left(x \right)} - 2\right)^{2} \log{\left(x \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2*log(x)      /   2         \ /2   2*log(x)\
  -2 + --------      \log (x) - 2*x/*|- - --------|
          x                          \x      x    /
------------------ + ------------------------------
   2                                         2     
log (x) - 2*log(x)       /   2              \      
                         \log (x) - 2*log(x)/

$$\frac{-2 + \frac{2 \log{\left(x \right)}}{x}}{\log{\left(x \right)}^{2} - 2 \log{\left(x \right)}} + \frac{\left(- 2 x + \log{\left(x \right)}^{2}\right) \left(- \frac{2 \log{\left(x \right)}}{x} + \frac{2}{x}\right)}{\left(\log{\left(x \right)}^{2} - 2 \log{\left(x \right)}\right)^{2}}$$

Simplify

The second derivative [src]

  /                /                      2  \                                                 \
  |                |       4*(-1 + log(x))   | /     2         \                               |
  |                |1 + ---------------------|*\- log (x) + 2*x/     /    log(x)\              |
  |                |                 2       |                     4*|1 - ------|*(-1 + log(x))|
  |  -1 + log(x)   \    (-2 + log(x)) *log(x)/                       \      x   /              |
2*|- ----------- - --------------------------------------------- + ----------------------------|
  \       x                           x*log(x)                         (-2 + log(x))*log(x)    /
------------------------------------------------------------------------------------------------
                                     x*(-2 + log(x))*log(x)

$$\frac{2 \left(\frac{4 \left(1 - \frac{\log{\left(x \right)}}{x}\right) \left(\log{\left(x \right)} - 1\right)}{\left(\log{\left(x \right)} - 2\right) \log{\left(x \right)}} - \frac{\left(1 + \frac{4 \left(\log{\left(x \right)} - 1\right)^{2}}{\left(\log{\left(x \right)} - 2\right)^{2} \log{\left(x \right)}}\right) \left(2 x - \log{\left(x \right)}^{2}\right)}{x \log{\left(x \right)}} - \frac{\log{\left(x \right)} - 1}{x}\right)}{x \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)}}$$

Simplify

The third derivative [src]

  /                               /                      2  \                                              /                                                     3   \\
  |                  /    log(x)\ |       4*(-1 + log(x))   |                            /     2         \ |                12*(-1 + log(x))     24*(-1 + log(x))    ||
  |                6*|1 - ------|*|1 + ---------------------|                            \- log (x) + 2*x/*|-5 + 2*log(x) + ---------------- + ----------------------||
  |                  \      x   / |                 2       |                     2                        |                     log(x)                     2    2   ||
  |-3 + 2*log(x)                  \    (-2 + log(x)) *log(x)/      6*(-1 + log(x))                         \                                   (-2 + log(x)) *log (x)/|
2*|------------- - ------------------------------------------ + ---------------------- + -----------------------------------------------------------------------------|
  \      x                           log(x)                     x*(-2 + log(x))*log(x)                               x*(-2 + log(x))*log(x)                           /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                         2                                                                                             
                                                                        x *(-2 + log(x))*log(x)

$$\frac{2 \left(- \frac{6 \left(1 - \frac{\log{\left(x \right)}}{x}\right) \left(1 + \frac{4 \left(\log{\left(x \right)} - 1\right)^{2}}{\left(\log{\left(x \right)} - 2\right)^{2} \log{\left(x \right)}}\right)}{\log{\left(x \right)}} + \frac{\left(2 x - \log{\left(x \right)}^{2}\right) \left(\frac{12 \left(\log{\left(x \right)} - 1\right)}{\log{\left(x \right)}} + 2 \log{\left(x \right)} - 5 + \frac{24 \left(\log{\left(x \right)} - 1\right)^{3}}{\left(\log{\left(x \right)} - 2\right)^{2} \log{\left(x \right)}^{2}}\right)}{x \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)}} + \frac{2 \log{\left(x \right)} - 3}{x} + \frac{6 \left(\log{\left(x \right)} - 1\right)^{2}}{x \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)}}\right)}{x^{2} \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)}}$$

Simplify

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Derivative of (lnx^2-2x)/(lnx^2-2lnx)

The graph:

Piecewise:

The solution