Derivative of √(x+√x)

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   ___________
  /       ___ 
\/  x + \/ x

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

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

$$\frac{d}{d x} \sqrt{\sqrt{x} + x}$$

Transform

Detail solution

Let $u = \sqrt{x} + x$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sqrt{x} + x\right)$ :
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of √(x+√x)

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The solution