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

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Derivative of:
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Identical expressions
one /(log(x)^(two))*x
1 divide by ( logarithm of (x) to the power of (2)) multiply by x
one divide by ( logarithm of (x) to the power of (two)) multiply by x
1/(log(x)(2))*x
1/logx2*x
1/(log(x)^(2))x
1/(log(x)(2))x
1/logx2x
1/logx^2x
1 divide by (log(x)^(2))*x

The derivative of the function/ 1/(log(x)^(2))*x

Derivative of 1/(log(x)^(2))*x

The solution

You have entered [src]

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

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

x/log(x)^2

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       12  \
2*|1 - -------|
  |       2   |
  \    log (x)/
---------------
    2    3     
   x *log (x)

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

Simplify

The graph

Derivative of 1/(log(x)^(2))*x