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Identical expressions
ln(sqrt(2x- one)+sqrt(2x))
ln( square root of (2x minus 1) plus square root of (2x))
ln( square root of (2x minus one) plus square root of (2x))
ln(√(2x-1)+√(2x))
lnsqrt2x-1+sqrt2x
Similar expressions
ln(sqrt(2x+1)+sqrt(2x))
ln(sqrt(2x-1)-sqrt(2x))

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

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

The solution

You have entered [src]

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

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

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

Detail solution

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

$\frac{\frac{1}{\sqrt{2 x - 1}} + \frac{\sqrt{2}}{2 \sqrt{x}}}{\sqrt{2 x} + \sqrt{2 x - 1}}$
Now simplify:

$\frac{\sqrt{x} + \frac{\sqrt{4 x - 2}}{2}}{2 x^{\frac{3}{2}} - \sqrt{x} + \sqrt{2} x \sqrt{2 x - 1}}$

The answer is:

$\frac{\sqrt{x} + \frac{\sqrt{4 x - 2}}{2}}{2 x^{\frac{3}{2}} - \sqrt{x} + \sqrt{2} x \sqrt{2 x - 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                ___   ___
     1        \/ 2 *\/ x 
----------- + -----------
  _________       2*x    
\/ 2*x - 1               
-------------------------
    _________     _____  
  \/ 2*x - 1  + \/ 2*x

$$\frac{\frac{1}{\sqrt{2 x - 1}} + \frac{\sqrt{2} \sqrt{x}}{2 x}}{\sqrt{2 x} + \sqrt{2 x - 1}}$$

Simplify

The second derivative [src]

 /                                                  2    \ 
 |                            /                 ___\     | 
 |                            |     2         \/ 2 |     | 
 |                            |------------ + -----|     | 
 |                  ___       |  __________     ___|     | 
 |      1         \/ 2        \\/ -1 + 2*x    \/ x /     | 
-|------------- + ------ + ------------------------------| 
 |          3/2      3/2     /  __________     ___   ___\| 
 \(-1 + 2*x)      4*x      4*\\/ -1 + 2*x  + \/ 2 *\/ x // 
-----------------------------------------------------------
                   __________     ___   ___                
                 \/ -1 + 2*x  + \/ 2 *\/ x

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

Simplify

The third derivative [src]

                                          3                                                                 
                    /                 ___\                    /                 ___\ /                  ___\
                    |     2         \/ 2 |                    |     2         \/ 2 | |      4         \/ 2 |
                    |------------ + -----|                  3*|------------ + -----|*|------------- + -----|
                    |  __________     ___|            ___     |  __________     ___| |          3/2     3/2|
      3             \\/ -1 + 2*x    \/ x /        3*\/ 2      \\/ -1 + 2*x    \/ x / \(-1 + 2*x)       x   /
------------- + ------------------------------- + ------- + ------------------------------------------------
          5/2                                 2       5/2              /  __________     ___   ___\         
(-1 + 2*x)        /  __________     ___   ___\     8*x               8*\\/ -1 + 2*x  + \/ 2 *\/ x /         
                4*\\/ -1 + 2*x  + \/ 2 *\/ x /                                                              
------------------------------------------------------------------------------------------------------------
                                           __________     ___   ___                                         
                                         \/ -1 + 2*x  + \/ 2 *\/ x

$$\frac{\frac{3 \left(\frac{4}{\left(2 x - 1\right)^{\frac{3}{2}}} + \frac{\sqrt{2}}{x^{\frac{3}{2}}}\right) \left(\frac{2}{\sqrt{2 x - 1}} + \frac{\sqrt{2}}{\sqrt{x}}\right)}{8 \left(\sqrt{2} \sqrt{x} + \sqrt{2 x - 1}\right)} + \frac{\left(\frac{2}{\sqrt{2 x - 1}} + \frac{\sqrt{2}}{\sqrt{x}}\right)^{3}}{4 \left(\sqrt{2} \sqrt{x} + \sqrt{2 x - 1}\right)^{2}} + \frac{3}{\left(2 x - 1\right)^{\frac{5}{2}}} + \frac{3 \sqrt{2}}{8 x^{\frac{5}{2}}}}{\sqrt{2} \sqrt{x} + \sqrt{2 x - 1}}$$

Simplify

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

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The solution