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

The solution

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\/   x + \/  x + \/ x

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

sqrt(x + sqrt(x + sqrt(x)))

Detail solution

Let $u = x + \sqrt{\sqrt{x} + x}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
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:
      1. Apply the power rule: $x$ goes to $1$
      2. 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:

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

$\frac{4 \sqrt{x} \sqrt{\sqrt{x} + x} + 2 \sqrt{x} + 1}{8 \sqrt{x} \sqrt{\sqrt{x} + x} \sqrt{x + \sqrt{\sqrt{x} + x}}}$

The answer is:

$\frac{4 \sqrt{x} \sqrt{\sqrt{x} + x} + 2 \sqrt{x} + 1}{8 \sqrt{x} \sqrt{\sqrt{x} + x} \sqrt{x + \sqrt{\sqrt{x} + x}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

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   /|
  |\    \/  x + \/ x  /      \       \x + \/ x /                    /     \    \/  x + \/ x  /                      |
3*|--------------------- + ------------------------------------------ + --------------------------------------------|
  |                    2                    ___________                        ___________ /       ___________\     |
  |/       ___________\                    /       ___                        /       ___  |      /       ___ |     |
  ||      /       ___ |                  \/  x + \/ x                       \/  x + \/ x  *\x + \/  x + \/ x  /     |
  \\x + \/  x + \/ x  /                                                                                             /
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                                             512*\/   x + \/  x + \/ x

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

Simplify

The graph

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

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