Derivative of 2x/ln(x)

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

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

(2*x)/log(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $2$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       6   \
2*|1 - -------|
  |       2   |
  \    log (x)/
---------------
    2    2     
   x *log (x)

$$\frac{2 \left(1 - \frac{6}{\log{\left(x \right)}^{2}}\right)}{x^{2} \log{\left(x \right)}^{2}}$$

Simplify

The graph