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Derivative of log(x)^2/x

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

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

log(x)^2/x

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $\frac{2 \log{\left(x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /          2               \
2*\-6 - 3*log (x) + 11*log(x)/
------------------------------
               4              
              x

$$\frac{2 \left(- 3 \log{\left(x \right)}^{2} + 11 \log{\left(x \right)} - 6\right)}{x^{4}}$$

Simplify

The graph