Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of cot(x)*cos(x)
Derivative of z
Derivative of 2x^3
Derivative of 1/x^5
Identical expressions
ln((sqrt(2x+ one)- one))/(sqrt(2x+ one)+ one)
ln(( square root of (2x plus 1) minus 1)) divide by ( square root of (2x plus 1) plus 1)
ln(( square root of (2x plus one) minus one)) divide by ( square root of (2x plus one) plus one)
ln((√(2x+1)-1))/(√(2x+1)+1)
lnsqrt2x+1-1/sqrt2x+1+1
ln((sqrt(2x+1)-1)) divide by (sqrt(2x+1)+1)
Similar expressions
ln((sqrt(2x+1)-1))/(sqrt(2x-1)+1)
ln((sqrt(2x-1)-1))/(sqrt(2x+1)+1)
ln((sqrt(2x+1)+1))/(sqrt(2x+1)+1)
ln((sqrt(2x+1)-1))/(sqrt(2x+1)-1)

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

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

The solution

You have entered [src]

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

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

  /   /  _________    \\
d |log\\/ 2*x + 1  - 1/|
--|--------------------|
dx|    _________       |
  \  \/ 2*x + 1  + 1   /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     3                        2                                 3                   /     1                      2                                2               \    /       _________\                                                                                                      
------------ + -------------------------------- + -----------------------------   3*|------------ + ---------------------------- + -------------------------------|*log\-1 + \/ 1 + 2*x /     /     1                      1              \       /     1                      2             \ 
         5/2                                  2            2 /       _________\     |         5/2   /      _________\          2                    2             |                         3*|------------ + ----------------------------|     3*|------------ + ---------------------------| 
(1 + 2*x)               3/2 /       _________\    (1 + 2*x) *\-1 + \/ 1 + 2*x /     |(1 + 2*x)      \1 + \/ 1 + 2*x /*(1 + 2*x)    /      _________\           3/2|                           |         3/2             /       _________\|       |         3/2   /      _________\          | 
               (1 + 2*x)   *\-1 + \/ 1 + 2*x /                                      \                                              \1 + \/ 1 + 2*x / *(1 + 2*x)   /                           \(1 + 2*x)      (1 + 2*x)*\-1 + \/ 1 + 2*x //       \(1 + 2*x)      \1 + \/ 1 + 2*x /*(1 + 2*x)/ 
------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------- + ------------------------------------------------ + ------------------------------------------------
                                       _________                                                                                    _________                                               /      _________\   _________ /       _________\   /      _________\   _________ /       _________\
                                -1 + \/ 1 + 2*x                                                                               1 + \/ 1 + 2*x                                                \1 + \/ 1 + 2*x /*\/ 1 + 2*x *\-1 + \/ 1 + 2*x /   \1 + \/ 1 + 2*x /*\/ 1 + 2*x *\-1 + \/ 1 + 2*x /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                              _________                                                                                                                                        
                                                                                                                                        1 + \/ 1 + 2*x

$$\frac{- \frac{3 \cdot \left(\frac{2}{\left(2 x + 1\right)^{2} \left(\sqrt{2 x + 1} + 1\right)} + \frac{2}{\left(2 x + 1\right)^{\frac{3}{2}} \left(\sqrt{2 x + 1} + 1\right)^{2}} + \frac{1}{\left(2 x + 1\right)^{\frac{5}{2}}}\right) \log{\left(\sqrt{2 x + 1} - 1 \right)}}{\sqrt{2 x + 1} + 1} + \frac{\frac{3}{\left(2 x + 1\right)^{2} \left(\sqrt{2 x + 1} - 1\right)} + \frac{2}{\left(2 x + 1\right)^{\frac{3}{2}} \left(\sqrt{2 x + 1} - 1\right)^{2}} + \frac{3}{\left(2 x + 1\right)^{\frac{5}{2}}}}{\sqrt{2 x + 1} - 1} + \frac{3 \cdot \left(\frac{1}{\left(2 x + 1\right) \left(\sqrt{2 x + 1} - 1\right)} + \frac{1}{\left(2 x + 1\right)^{\frac{3}{2}}}\right)}{\sqrt{2 x + 1} \left(\sqrt{2 x + 1} - 1\right) \left(\sqrt{2 x + 1} + 1\right)} + \frac{3 \cdot \left(\frac{2}{\left(2 x + 1\right) \left(\sqrt{2 x + 1} + 1\right)} + \frac{1}{\left(2 x + 1\right)^{\frac{3}{2}}}\right)}{\sqrt{2 x + 1} \left(\sqrt{2 x + 1} - 1\right) \left(\sqrt{2 x + 1} + 1\right)}}{\sqrt{2 x + 1} + 1}$$

Simplify

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution