Derivative of sqrt(x+sqrt(x+sqrt(x)))-sqrt(x)

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    ____________________        
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\/   x + \/  x + \/ x    - \/ x

$$- \sqrt{x} + \sqrt{x + \sqrt{\sqrt{x} + x}}$$

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d |  /        /       ___       ___|
--\\/   x + \/  x + \/ x    - \/ x /
dx

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

Transform

Detail solution

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                    1      1        
                    - + -------     
                    2       ___     
              1         4*\/ x      
              - + ----------------  
              2        ___________  
                      /       ___   
     1            2*\/  x + \/ x    
- ------- + ------------------------
      ___       ____________________
  2*\/ x       /        ___________ 
              /        /       ___  
            \/   x + \/  x + \/ x

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

Simplify

The second derivative [src]

                            2                                           
        /            1     \               /                  2\        
        |      2 + -----   |               |       /      1  \ |        
        |            ___   |               |       |2 + -----| |        
        |          \/ x    |               |       |      ___| |        
        |4 + --------------|               | 2     \    \/ x / |        
        |       ___________|             2*|---- + ------------|        
        |      /       ___ |               | 3/2          ___  |        
 16     \    \/  x + \/ x  /               \x       x + \/ x   /        
---- - ----------------------- - ---------------------------------------
 3/2                       3/2                      ____________________
x      /       ___________\         ___________    /        ___________ 
       |      /       ___ |        /       ___    /        /       ___  
       \x + \/  x + \/ x  /      \/  x + \/ x  *\/   x + \/  x + \/ x   
------------------------------------------------------------------------
                                   64

$$\frac{- \frac{\left(\frac{2 + \frac{1}{\sqrt{x}}}{\sqrt{\sqrt{x} + x}} + 4\right)^{2}}{\left(x + \sqrt{\sqrt{x} + x}\right)^{\frac{3}{2}}} - \frac{2 \left(\frac{\left(2 + \frac{1}{\sqrt{x}}\right)^{2}}{\sqrt{x} + x} + \frac{2}{x^{\frac{3}{2}}}\right)}{\sqrt{\sqrt{x} + x} \sqrt{x + \sqrt{\sqrt{x} + x}}} + \frac{16}{x^{\frac{3}{2}}}}{64}$$

Simplify

The third derivative [src]

  /                              3      /                  3                   \                          /                  2\\
  |          /            1     \       |       /      1  \       /      1  \  |     /            1     \ |       /      1  \ ||
  |          |      2 + -----   |       |       |2 + -----|     2*|2 + -----|  |     |      2 + -----   | |       |2 + -----| ||
  |          |            ___   |       |       |      ___|       |      ___|  |     |            ___   | |       |      ___| ||
  |          |          \/ x    |       | 4     \    \/ x /       \    \/ x /  |     |          \/ x    | | 2     \    \/ x / ||
  |          |4 + --------------|     4*|---- + ------------ + ----------------|   2*|4 + --------------|*|---- + ------------||
  |          |       ___________|       | 5/2              2    3/2 /      ___\|     |       ___________| | 3/2          ___  ||
  |          |      /       ___ |       |x      /      ___\    x   *\x + \/ x /|     |      /       ___ | \x       x + \/ x   /|
  |   64     \    \/  x + \/ x  /       \       \x + \/ x /                    /     \    \/  x + \/ x  /                      |
3*|- ---- + ----------------------- + ------------------------------------------ + --------------------------------------------|
  |   5/2                       5/2                       ____________________                                           3/2   |
  |  x      /       ___________\          ___________    /        ___________            ___________ /       ___________\      |
  |         |      /       ___ |         /       ___    /        /       ___            /       ___  |      /       ___ |      |
  \         \x + \/  x + \/ x  /       \/  x + \/ x  *\/   x + \/  x + \/ x           \/  x + \/ x  *\x + \/  x + \/ x  /      /
--------------------------------------------------------------------------------------------------------------------------------
                                                              512

$$\frac{3 \left(\frac{\left(\frac{2 + \frac{1}{\sqrt{x}}}{\sqrt{\sqrt{x} + x}} + 4\right)^{3}}{\left(x + \sqrt{\sqrt{x} + x}\right)^{\frac{5}{2}}} + \frac{4 \left(\frac{\left(2 + \frac{1}{\sqrt{x}}\right)^{3}}{\left(\sqrt{x} + x\right)^{2}} + \frac{2 \cdot \left(2 + \frac{1}{\sqrt{x}}\right)}{x^{\frac{3}{2}} \left(\sqrt{x} + x\right)} + \frac{4}{x^{\frac{5}{2}}}\right)}{\sqrt{\sqrt{x} + x} \sqrt{x + \sqrt{\sqrt{x} + x}}} + \frac{2 \left(\frac{2 + \frac{1}{\sqrt{x}}}{\sqrt{\sqrt{x} + x}} + 4\right) \left(\frac{\left(2 + \frac{1}{\sqrt{x}}\right)^{2}}{\sqrt{x} + x} + \frac{2}{x^{\frac{3}{2}}}\right)}{\sqrt{\sqrt{x} + x} \left(x + \sqrt{\sqrt{x} + x}\right)^{\frac{3}{2}}} - \frac{64}{x^{\frac{5}{2}}}\right)}{512}$$

Simplify

The graph

Derivative of sqrt(x+sqrt(x+sqrt(x)))-sqrt(x)

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Derivative of sqrt(x+sqrt(x+sqrt(x)))-sqrt(x)

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