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

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

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

The solution

You have entered [src]

           1         
1*-------------------
    _______     _____
  \/ x + 2  - \/ 2*x

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

d /           1         \
--|1*-------------------|
dx|    _______     _____|
  \  \/ x + 2  - \/ 2*x /

$$\frac{d}{d x} 1 \cdot \frac{1}{- \sqrt{2 x} + \sqrt{x + 2}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 1$ and $g{\left(x \right)} = - \sqrt{2} \sqrt{x} + \sqrt{x + 2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $- \sqrt{2} \sqrt{x} + \sqrt{x + 2}$ term by term:
  1. Let $u = x + 2$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 2\right)$ :
    1. Differentiate $x + 2$ term by term:
      1. The derivative of the constant $2$ is zero.
      2. Apply the power rule: $x$ goes to $1$
      The result is: $1$
    The result of the chain rule is:
    
    $\frac{1}{2 \sqrt{x + 2}}$
  4. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    So, the result is: $- \frac{\sqrt{2}}{2 \sqrt{x}}$
  The result is: $\frac{1}{2 \sqrt{x + 2}} - \frac{\sqrt{2}}{2 \sqrt{x}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                   ___ 
       1         \/ 2  
- ----------- + -------
      _______       ___
  2*\/ x + 2    2*\/ x 
-----------------------
                      2
 /  _______     _____\ 
 \\/ x + 2  - \/ 2*x /

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

Simplify

The second derivative [src]

 /                                              2 \ 
 |                           /              ___\  | 
 |                           |    1       \/ 2 |  | 
 |                         2*|--------- - -----|  | 
 |                 ___       |  _______     ___|  | 
 |      1        \/ 2        \\/ 2 + x    \/ x /  | 
-|- ---------- + ----- + -------------------------| 
 |         3/2     3/2       _______     ___   ___| 
 \  (2 + x)       x      - \/ 2 + x  + \/ 2 *\/ x / 
----------------------------------------------------
                                        2           
             /    _______     ___   ___\            
           4*\- \/ 2 + x  + \/ 2 *\/ x /

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

Simplify

The third derivative [src]

  /                                               3                                                \
  |                            /              ___\         /               ___\ /              ___\|
  |                            |    1       \/ 2 |         |    1        \/ 2 | |    1       \/ 2 ||
  |                          2*|--------- - -----|       2*|---------- - -----|*|--------- - -----||
  |                 ___        |  _______     ___|         |       3/2     3/2| |  _______     ___||
  |      1        \/ 2         \\/ 2 + x    \/ x /         \(2 + x)       x   / \\/ 2 + x    \/ x /|
3*|- ---------- + ----- - ---------------------------- + ------------------------------------------|
  |         5/2     5/2                              2               _______     ___   ___         |
  |  (2 + x)       x      /    _______     ___   ___\            - \/ 2 + x  + \/ 2 *\/ x          |
  \                       \- \/ 2 + x  + \/ 2 *\/ x /                                              /
----------------------------------------------------------------------------------------------------
                                                                2                                   
                                     /    _______     ___   ___\                                    
                                   8*\- \/ 2 + x  + \/ 2 *\/ x /

$$\frac{3 \cdot \left(\frac{2 \left(\frac{1}{\left(x + 2\right)^{\frac{3}{2}}} - \frac{\sqrt{2}}{x^{\frac{3}{2}}}\right) \left(\frac{1}{\sqrt{x + 2}} - \frac{\sqrt{2}}{\sqrt{x}}\right)}{\sqrt{2} \sqrt{x} - \sqrt{x + 2}} - \frac{2 \left(\frac{1}{\sqrt{x + 2}} - \frac{\sqrt{2}}{\sqrt{x}}\right)^{3}}{\left(\sqrt{2} \sqrt{x} - \sqrt{x + 2}\right)^{2}} - \frac{1}{\left(x + 2\right)^{\frac{5}{2}}} + \frac{\sqrt{2}}{x^{\frac{5}{2}}}\right)}{8 \left(\sqrt{2} \sqrt{x} - \sqrt{x + 2}\right)^{2}}$$

Simplify

The graph

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Identical expressions

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

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The solution