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(lnx-1)/(lnx)^2

Derivative of (lnx-1)/(lnx)^2

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(x) - 1
----------
    2     
 log (x)  
$$\frac{\log{\left(x \right)} - 1}{\log{\left(x \right)}^{2}}$$
(log(x) - 1)/log(x)^2
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. The derivative of is .

      The result is:

    To find :

    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:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
    1       2*(log(x) - 1)
--------- - --------------
     2             3      
x*log (x)     x*log (x)   
$$- \frac{2 \left(\log{\left(x \right)} - 1\right)}{x \log{\left(x \right)}^{3}} + \frac{1}{x \log{\left(x \right)}^{2}}$$
The second derivative [src]
                /      3   \              
              2*|1 + ------|*(-1 + log(x))
       4        \    log(x)/              
-1 - ------ + ----------------------------
     log(x)              log(x)           
------------------------------------------
                 2    2                   
                x *log (x)                
$$\frac{\frac{2 \left(1 + \frac{3}{\log{\left(x \right)}}\right) \left(\log{\left(x \right)} - 1\right)}{\log{\left(x \right)}} - 1 - \frac{4}{\log{\left(x \right)}}}{x^{2} \log{\left(x \right)}^{2}}$$
The third derivative [src]
  /                                            /      9         12  \\
  |               /      3   \   (-1 + log(x))*|2 + ------ + -------||
  |             3*|1 + ------|                 |    log(x)      2   ||
  |      3        \    log(x)/                 \             log (x)/|
2*|1 + ------ + -------------- - ------------------------------------|
  \    log(x)       log(x)                      log(x)               /
----------------------------------------------------------------------
                               3    2                                 
                              x *log (x)                              
$$\frac{2 \left(\frac{3 \left(1 + \frac{3}{\log{\left(x \right)}}\right)}{\log{\left(x \right)}} - \frac{\left(\log{\left(x \right)} - 1\right) \left(2 + \frac{9}{\log{\left(x \right)}} + \frac{12}{\log{\left(x \right)}^{2}}\right)}{\log{\left(x \right)}} + 1 + \frac{3}{\log{\left(x \right)}}\right)}{x^{3} \log{\left(x \right)}^{2}}$$
The graph
Derivative of (lnx-1)/(lnx)^2