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Identical expressions
(lnx- one)/(lnx)^ two
(lnx minus 1) divide by (lnx) squared
(lnx minus one) divide by (lnx) to the power of two
(lnx-1)/(lnx)2
lnx-1/lnx2
(lnx-1)/(lnx)²
(lnx-1)/(lnx) to the power of 2
lnx-1/lnx^2
(lnx-1) divide by (lnx)^2
Similar expressions
(lnx+1)/(lnx)^2

The derivative of the function/ (lnx-1)/(lnx)^2

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

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

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(x \right)} - 1$ and $g{\left(x \right)} = \log{\left(x \right)}^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\log{\left(x \right)} - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\frac{1}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
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:
  
  $\frac{2 \log{\left(x \right)}}{x}$
Now plug in to the quotient rule:

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

$\frac{2 - \log{\left(x \right)}}{x \log{\left(x \right)}^{3}}$

The answer is:

$\frac{2 - \log{\left(x \right)}}{x \log{\left(x \right)}^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1       2*(log(x) - 1)
--------- - --------------
     2             3      
x*log (x)     x*log (x)

$$- \frac{2 \left(\log{\left(x \right)} - 1\right)}{x \log{\left(x \right)}^{3}} + \frac{1}{x \log{\left(x \right)}^{2}}$$

Simplify

The second derivative [src]

                /      3   \              
              2*|1 + ------|*(-1 + log(x))
       4        \    log(x)/              
-1 - ------ + ----------------------------
     log(x)              log(x)           
------------------------------------------
                 2    2                   
                x *log (x)

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

Simplify

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)

$$\frac{2 \left(\frac{3 \left(1 + \frac{3}{\log{\left(x \right)}}\right)}{\log{\left(x \right)}} - \frac{\left(\log{\left(x \right)} - 1\right) \left(2 + \frac{9}{\log{\left(x \right)}} + \frac{12}{\log{\left(x \right)}^{2}}\right)}{\log{\left(x \right)}} + 1 + \frac{3}{\log{\left(x \right)}}\right)}{x^{3} \log{\left(x \right)}^{2}}$$

Simplify

The graph

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

The graph:

Piecewise:

The solution