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Identical expressions
ln(sqrt(e^x+ one)- one)
ln( square root of (e to the power of x plus 1) minus 1)
ln( square root of (e to the power of x plus one) minus one)
ln(√(e^x+1)-1)
ln(sqrt(ex+1)-1)
lnsqrtex+1-1
lnsqrte^x+1-1
Similar expressions
ln(sqrt(e^x-1)-1)
ln(sqrt(e^x+1)+1)

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

Derivative of ln(sqrt(e^x+1)-1)

The solution

You have entered [src]

   /   ________    \
   |  /  x         |
log\\/  E  + 1  - 1/

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

log(sqrt(E^x + 1) - 1)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                x              
               e               
-------------------------------
     ________ /   ________    \
    /  x      |  /  x         |
2*\/  E  + 1 *\\/  E  + 1  - 1/

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

Simplify

The second derivative [src]

/                    x                     x            \   
|     2             e                     e             |  x
|----------- - ----------- - ---------------------------|*e 
|   ________           3/2            /        ________\|   
|  /      x    /     x\      /     x\ |       /      x ||   
\\/  1 + e     \1 + e /      \1 + e /*\-1 + \/  1 + e  //   
------------------------------------------------------------
                      /        ________\                    
                      |       /      x |                    
                    4*\-1 + \/  1 + e  /

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

Simplify

The third derivative [src]

/                     x            2*x                    x                              2*x                             2*x           \   
|     4            6*e          3*e                    6*e                            2*e                             3*e              |  x
|----------- - ----------- + ----------- - --------------------------- + ------------------------------- + ----------------------------|*e 
|   ________           3/2           5/2            /        ________\                                 2           2 /        ________\|   
|  /      x    /     x\      /     x\      /     x\ |       /      x |           3/2 /        ________\    /     x\  |       /      x ||   
|\/  1 + e     \1 + e /      \1 + e /      \1 + e /*\-1 + \/  1 + e  /   /     x\    |       /      x |    \1 + e / *\-1 + \/  1 + e  /|   
\                                                                        \1 + e /   *\-1 + \/  1 + e  /                                /   
-------------------------------------------------------------------------------------------------------------------------------------------
                                                              /        ________\                                                           
                                                              |       /      x |                                                           
                                                            8*\-1 + \/  1 + e  /

$$\frac{\left(\frac{4}{\sqrt{e^{x} + 1}} - \frac{6 e^{x}}{\left(e^{x} + 1\right)^{\frac{3}{2}}} + \frac{3 e^{2 x}}{\left(e^{x} + 1\right)^{\frac{5}{2}}} - \frac{6 e^{x}}{\left(\sqrt{e^{x} + 1} - 1\right) \left(e^{x} + 1\right)} + \frac{3 e^{2 x}}{\left(\sqrt{e^{x} + 1} - 1\right) \left(e^{x} + 1\right)^{2}} + \frac{2 e^{2 x}}{\left(\sqrt{e^{x} + 1} - 1\right)^{2} \left(e^{x} + 1\right)^{\frac{3}{2}}}\right) e^{x}}{8 \left(\sqrt{e^{x} + 1} - 1\right)}$$

Simplify

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Derivative of ln(sqrt(e^x+1)-1)

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Piecewise:

The solution