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y=ln(one /(x+sqrt((x^ two)- one)))
y equally ln(1 divide by (x plus square root of ((x squared ) minus 1)))
y equally ln(one divide by (x plus square root of ((x to the power of two) minus one)))
y=ln(1/(x+√((x^2)-1)))
y=ln(1/(x+sqrt((x2)-1)))
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y=ln(1/(x+sqrt((x to the power of 2)-1)))
y=ln1/x+sqrtx^2-1
y=ln(1 divide by (x+sqrt((x^2)-1)))
Similar expressions
y=ln(1/(x-sqrt((x^2)-1)))
y=ln(1/(x+sqrt((x^2)+1)))

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

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

The solution

You have entered [src]

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

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

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

Detail solution

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

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The solution