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Identical expressions
log((one + two *x)/(one - two *x))
logarithm of ((1 plus 2 multiply by x) divide by (1 minus 2 multiply by x))
logarithm of ((one plus two multiply by x) divide by (one minus two multiply by x))
log((1+2x)/(1-2x))
log1+2x/1-2x
log((1+2*x) divide by (1-2*x))
Similar expressions
log((1+2*x)/(1+2*x))
log((1-2*x)/(1-2*x))

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

Derivative of log((1+2x)/(1-2x))

The solution

You have entered [src]

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

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

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

Detail solution

Let $u = \frac{2 x + 1}{1 - 2 x}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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.
      1. 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.
      1. 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:

$\frac{4}{\left(1 - 2 x\right) \left(2 x + 1\right)}$
Now simplify:

$- \frac{4}{4 x^{2} - 1}$

The answer is:

$- \frac{4}{4 x^{2} - 1}$

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /    1 + 2*x \ /     1         1    \
4*|1 - --------|*|- ------- - --------|
  \    -1 + 2*x/ \  1 + 2*x   -1 + 2*x/
---------------------------------------
                1 + 2*x

$$\frac{4 \left(1 - \frac{2 x + 1}{2 x - 1}\right) \left(- \frac{1}{2 x + 1} - \frac{1}{2 x - 1}\right)}{2 x + 1}$$

Simplify

3-я производная [src]

   /    1 + 2*x \ /    1             1                 1          \
16*|1 - --------|*|---------- + ----------- + --------------------|
   \    -1 + 2*x/ |         2             2   (1 + 2*x)*(-1 + 2*x)|
                  \(1 + 2*x)    (-1 + 2*x)                        /
-------------------------------------------------------------------
                              1 + 2*x

$$\frac{16 \left(1 - \frac{2 x + 1}{2 x - 1}\right) \left(\frac{1}{\left(2 x + 1\right)^{2}} + \frac{1}{\left(2 x - 1\right) \left(2 x + 1\right)} + \frac{1}{\left(2 x - 1\right)^{2}}\right)}{2 x + 1}$$

Simplify

The third derivative [src]

   /    1 + 2*x \ /    1             1                 1          \
16*|1 - --------|*|---------- + ----------- + --------------------|
   \    -1 + 2*x/ |         2             2   (1 + 2*x)*(-1 + 2*x)|
                  \(1 + 2*x)    (-1 + 2*x)                        /
-------------------------------------------------------------------
                              1 + 2*x

Simplify

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Derivative of log((1+2x)/(1-2x))

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of log((1+2*x)/(1-2*x))

The graph:

Piecewise:

The solution

Derivative of log((1+2x)/(1-2x))