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ln(x+sqrt(x^ two - one))
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Similar expressions
ln(x+sqrt(x^2+1))
ln(x-sqrt(x^2-1))

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

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

The solution

You have entered [src]

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

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

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

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         x     
1 + -----------
       ________
      /  2     
    \/  x  - 1 
---------------
       ________
      /  2     
x + \/  x  - 1

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

Simplify

The second derivative [src]

 /                  2               \ 
 |/         x      \             2  | 
 ||1 + ------------|            x   | 
 ||       _________|    -1 + -------| 
 ||      /       2 |               2| 
 |\    \/  -1 + x  /         -1 + x | 
-|------------------- + ------------| 
 |         _________       _________| 
 |        /       2       /       2 | 
 \  x + \/  -1 + x      \/  -1 + x  / 
--------------------------------------
                  _________           
                 /       2            
           x + \/  -1 + x

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

Simplify

The third derivative [src]

                    3                                             /         2  \
  /         x      \        /         2  \     /         x      \ |        x   |
2*|1 + ------------|        |        x   |   3*|1 + ------------|*|-1 + -------|
  |       _________|    3*x*|-1 + -------|     |       _________| |           2|
  |      /       2 |        |           2|     |      /       2 | \     -1 + x /
  \    \/  -1 + x  /        \     -1 + x /     \    \/  -1 + x  /               
--------------------- + ------------------ + -----------------------------------
                   2                3/2           _________ /       _________\  
 /       _________\        /      2\             /       2  |      /       2 |  
 |      /       2 |        \-1 + x /           \/  -1 + x  *\x + \/  -1 + x  /  
 \x + \/  -1 + x  /                                                             
--------------------------------------------------------------------------------
                                       _________                                
                                      /       2                                 
                                x + \/  -1 + x

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

Simplify

The graph

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

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The solution