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Identical expressions
y=lnsqrt(one +2x)/(one -2x)
y equally ln square root of (1 plus 2x) divide by (1 minus 2x)
y equally ln square root of (one plus 2x) divide by (one minus 2x)
y=ln√(1+2x)/(1-2x)
y=lnsqrt1+2x/1-2x
y=lnsqrt(1+2x) divide by (1-2x)
Similar expressions
y=lnsqrt(1-2x)/(1-2x)
y=lnsqrt(1+2x)/(1+2x)

The derivative of the function/ y=lnsqrt(1+2x)/(1-2x)

Derivative of y=lnsqrt(1+2x)/(1-2x)

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sqrt{2 x + 1}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{2 x + 1}$ :
  1. Let $u = 2 x + 1$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 1\right)$ :
    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: $2$
    The result of the chain rule is:
    
    $\frac{1}{\sqrt{2 x + 1}}$
  The result of the chain rule is:
  
  $\frac{1}{2 x + 1}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                                                     /  _________\\
  |      2                  6                       3             12*log\\/ 1 + 2*x /|
4*|- ---------- - --------------------- - --------------------- + -------------------|
  |           3                       2            2                            3    |
  \  (1 + 2*x)    (1 + 2*x)*(-1 + 2*x)    (1 + 2*x) *(-1 + 2*x)       (-1 + 2*x)     /
--------------------------------------------------------------------------------------
                                       -1 + 2*x

$$\frac{4 \left(- \frac{2}{\left(2 x + 1\right)^{3}} - \frac{3}{\left(2 x - 1\right) \left(2 x + 1\right)^{2}} - \frac{6}{\left(2 x - 1\right)^{2} \left(2 x + 1\right)} + \frac{12 \log{\left(\sqrt{2 x + 1} \right)}}{\left(2 x - 1\right)^{3}}\right)}{2 x - 1}$$

Simplify

The graph

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Derivative of y=lnsqrt(1+2x)/(1-2x)

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The solution