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Identical expressions
sqrt(log(x+ one /x))
square root of ( logarithm of (x plus 1 divide by x))
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√(log(x+1/x))
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Similar expressions
sqrt(log(x-1/x))

The derivative of the function/ sqrt(log(x+1/x))

Derivative of sqrt(log(x+1/x))

The solution

You have entered [src]

    ____________
   /    /    1\ 
  /  log|x + -| 
\/      \    x/

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

sqrt(log(x + 1/x))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              1           
          1 - --          
               2          
              x           
--------------------------
              ____________
  /    1\    /    /    1\ 
2*|x + -|*  /  log|x + -| 
  \    x/ \/      \    x/

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

Simplify

The second derivative [src]

             2                2      
     /    1 \         /    1 \       
     |1 - --|         |1 - --|       
     |     2|         |     2|       
1    \    x /         \    x /       
-- - --------- - --------------------
 3     /    1\     /    1\    /    1\
x    2*|x + -|   4*|x + -|*log|x + -|
       \    x/     \    x/    \    x/
-------------------------------------
                   ____________      
       /    1\    /    /    1\       
       |x + -|*  /  log|x + -|       
       \    x/ \/      \    x/

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

Simplify

The third derivative [src]

               3                               3                       3                                
       /    1 \      /    1 \          /    1 \                /    1 \                   /    1 \      
       |1 - --|    3*|1 - --|        3*|1 - --|              3*|1 - --|                 3*|1 - --|      
       |     2|      |     2|          |     2|                |     2|                   |     2|      
  3    \    x /      \    x /          \    x /                \    x /                   \    x /      
- -- + --------- - ---------- + --------------------- + ---------------------- - -----------------------
   4           2    3 /    1\            2                       2                  3 /    1\    /    1\
  x     /    1\    x *|x + -|     /    1\     /    1\     /    1\     2/    1\   2*x *|x + -|*log|x + -|
        |x + -|       \    x/   4*|x + -| *log|x + -|   8*|x + -| *log |x + -|        \    x/    \    x/
        \    x/                   \    x/     \    x/     \    x/      \    x/                          
--------------------------------------------------------------------------------------------------------
                                                    ____________                                        
                                        /    1\    /    /    1\                                         
                                        |x + -|*  /  log|x + -|                                         
                                        \    x/ \/      \    x/

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

Simplify

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Derivative of sqrt(log(x+1/x))

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The solution