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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

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
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Differentiate term by term:

      1. Let .

      2. Apply the power rule: goes to

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of is .

        The result of the chain rule is:

      4. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result is:

    To find :

    1. Differentiate term by term:

      1. Let .

      2. Apply the power rule: goes to

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of is .

        The result of the chain rule is:

      4. The derivative of a constant times a function is the constant times the derivative of the function.

        1. The derivative of is .

        So, the result is:

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

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