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y=1/(2x*logx+x)

The derivative of the function/ y=1/(2x*logx-x)

Derivative of y=1/(2x*logx-x)

The solution

You have entered [src]

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

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

d /        1       \
--|1*--------------|
dx\  2*x*log(x) - x/

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                    3                   \
  |    3*(1 + 2*log(x))    6*(1 + 2*log(x))|
2*|1 - ----------------- + ----------------|
  |                    2    -1 + 2*log(x)  |
  \     (-1 + 2*log(x))                    /
--------------------------------------------
             4                2             
            x *(-1 + 2*log(x))

$$\frac{2 \cdot \left(1 + \frac{6 \cdot \left(2 \log{\left(x \right)} + 1\right)}{2 \log{\left(x \right)} - 1} - \frac{3 \left(2 \log{\left(x \right)} + 1\right)^{3}}{\left(2 \log{\left(x \right)} - 1\right)^{2}}\right)}{x^{4} \left(2 \log{\left(x \right)} - 1\right)^{2}}$$

Simplify

The graph

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Derivative of y=1/(2x*logx-x)

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The solution