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Identical expressions
x*sqrt(x+(one /x)- two)
x multiply by square root of (x plus (1 divide by x) minus 2)
x multiply by square root of (x plus (one divide by x) minus two)
x*√(x+(1/x)-2)
xsqrt(x+(1/x)-2)
xsqrtx+1/x-2
x*sqrt(x+(1 divide by x)-2)
Similar expressions
x*sqrt(x-(1/x)-2)
x*sqrt(x+(1/x)+2)

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

Derivative of x*sqrt(x+(1/x)-2)

The solution

You have entered [src]

      ___________
     /     1     
x*  /  x + - - 2 
  \/       x

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

x*sqrt(x + 1/x - 2)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

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

$\frac{3 x^{2} - 4 x + 1}{2 x \sqrt{x - 2 + \frac{1}{x}}}$

The answer is:

$\frac{3 x^{2} - 4 x + 1}{2 x \sqrt{x - 2 + \frac{1}{x}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      /1    1  \ 
                    x*|- - ----| 
    ___________       |2      2| 
   /     1            \    2*x / 
  /  x + - - 2  + ---------------
\/       x            ___________
                     /     1     
                    /  x + - - 2 
                  \/       x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of x*sqrt(x+(1/x)-2)

The graph:

Piecewise:

The solution