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Identical expressions
ln((one +2x)/(one -2x))- three /x
ln((1 plus 2x) divide by (1 minus 2x)) minus 3 divide by x
ln((one plus 2x) divide by (one minus 2x)) minus three divide by x
ln1+2x/1-2x-3/x
ln((1+2x) divide by (1-2x))-3 divide by x
Similar expressions
ln((1+2x)/(1-2x))+3/x
ln((1-2x)/(1-2x))-3/x
ln((1+2x)/(1+2x))-3/x

The derivative of the function/ ln((1+2x)/(1-2x))-3/x

Derivative of ln((1+2x)/(1-2x))-3/x

The solution

You have entered [src]

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

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

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

Detail solution

Differentiate $\log{\left(\frac{2 x + 1}{1 - 2 x} \right)} - \frac{3}{x}$ term by term:
1. Let $u = \frac{2 x + 1}{1 - 2 x}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x + 1}{1 - 2 x}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 2 x + 1$ and $g{\left(x \right)} = 1 - 2 x$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Differentiate $2 x + 1$ term by term:
      1. The derivative of the constant $1$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      The result is: $2$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Differentiate $1 - 2 x$ term by term:
      1. The derivative of the constant $1$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-2$
      The result is: $-2$
    Now plug in to the quotient rule:
    
    $\frac{4}{\left(1 - 2 x\right)^{2}}$
  The result of the chain rule is:
  
  $\frac{4}{\left(1 - 2 x\right) \left(2 x + 1\right)}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
  So, the result is: $\frac{3}{x^{2}}$
The result is: $\frac{4}{\left(1 - 2 x\right) \left(2 x + 1\right)} + \frac{3}{x^{2}}$
Now simplify:

$\frac{8 x^{2} - 3}{4 x^{4} - x^{2}}$

The answer is:

$\frac{8 x^{2} - 3}{4 x^{4} - x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       /    1 + 2*x \        /    1 + 2*x \          /    1 + 2*x \  \
  |     8*|1 - --------|      8*|1 - --------|        8*|1 - --------|  |
  |9      \    -1 + 2*x/        \    -1 + 2*x/          \    -1 + 2*x/  |
2*|-- + ---------------- + --------------------- + ---------------------|
  | 4               3                          2            2           |
  \x       (1 + 2*x)       (1 + 2*x)*(-1 + 2*x)    (1 + 2*x) *(-1 + 2*x)/

$$2 \left(\frac{8 \left(1 - \frac{2 x + 1}{2 x - 1}\right)}{\left(2 x + 1\right)^{3}} + \frac{8 \left(1 - \frac{2 x + 1}{2 x - 1}\right)}{\left(2 x - 1\right) \left(2 x + 1\right)^{2}} + \frac{8 \left(1 - \frac{2 x + 1}{2 x - 1}\right)}{\left(2 x - 1\right)^{2} \left(2 x + 1\right)} + \frac{9}{x^{4}}\right)$$

Simplify

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Derivative of ln((1+2x)/(1-2x))-3/x

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The solution