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

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

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

The solution

You have entered [src]

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

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

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

Detail solution

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

$\frac{- \frac{1}{x^{2}} - \frac{1}{x^{3} \sqrt{1 + \frac{1}{x^{2}}}}}{\sqrt{1 + \frac{1}{x^{2}}} + \frac{1}{x}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1           1        
- -- - ----------------
   2           ________
  x     3     / 1      
       x *   /  -- + 1 
            /    2     
          \/    x      
-----------------------
            ________   
   1       / 1         
   - +    /  -- + 1    
   x     /    2        
       \/    x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                                3                                                                 
                                                           /           1       \      /           1       \ /          1                 3       \
                                                         2*|1 + ---------------|    3*|1 + ---------------|*|2 - -------------- + ---------------|
                                                           |           ________|      |           ________| |               3/2          ________|
                                                           |          /     1  |      |          /     1  | |     3 /    1 \            /     1  |
                                                           |    x*   /  1 + -- |      |    x*   /  1 + -- | |    x *|1 + --|      x*   /  1 + -- |
                                                           |        /        2 |      |        /        2 | |       |     2|          /        2 |
            12               3                9            \      \/        x  /      \      \/        x  / \       \    x /        \/        x  /
-6 - --------------- - -------------- + -------------- - ------------------------ + --------------------------------------------------------------
            ________              5/2              3/2                         2                          /         ________\                     
           /     1      5 /    1 \       3 /    1 \         /         ________\                           |1       /     1  |                     
     x*   /  1 + --    x *|1 + --|      x *|1 + --|       2 |1       /     1  |                         x*|- +    /  1 + -- |                     
         /        2       |     2|         |     2|      x *|- +    /  1 + -- |                           |x     /        2 |                     
       \/        x        \    x /         \    x /         |x     /        2 |                           \    \/        x  /                     
                                                            \    \/        x  /                                                                   
--------------------------------------------------------------------------------------------------------------------------------------------------
                                                                 /         ________\                                                              
                                                               4 |1       /     1  |                                                              
                                                              x *|- +    /  1 + -- |                                                              
                                                                 |x     /        2 |                                                              
                                                                 \    \/        x  /

$$\frac{-6 + \frac{3 \left(1 + \frac{1}{x \sqrt{1 + \frac{1}{x^{2}}}}\right) \left(2 + \frac{3}{x \sqrt{1 + \frac{1}{x^{2}}}} - \frac{1}{x^{3} \left(1 + \frac{1}{x^{2}}\right)^{\frac{3}{2}}}\right)}{x \left(\sqrt{1 + \frac{1}{x^{2}}} + \frac{1}{x}\right)} - \frac{12}{x \sqrt{1 + \frac{1}{x^{2}}}} - \frac{2 \left(1 + \frac{1}{x \sqrt{1 + \frac{1}{x^{2}}}}\right)^{3}}{x^{2} \left(\sqrt{1 + \frac{1}{x^{2}}} + \frac{1}{x}\right)^{2}} + \frac{9}{x^{3} \left(1 + \frac{1}{x^{2}}\right)^{\frac{3}{2}}} - \frac{3}{x^{5} \left(1 + \frac{1}{x^{2}}\right)^{\frac{5}{2}}}}{x^{4} \left(\sqrt{1 + \frac{1}{x^{2}}} + \frac{1}{x}\right)}$$

Simplify

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

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The solution