Derivative of y=sqrt(x+2*(sqrt(x)))

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   _____________
  /         ___ 
\/  x + 2*\/ x

$$\sqrt{2 \sqrt{x} + x}$$

sqrt(x + 2*sqrt(x))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1      1      
  - + -------   
  2       ___   
      2*\/ x    
----------------
   _____________
  /         ___ 
\/  x + 2*\/ x

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

Simplify

The second derivative [src]

 /                  2\ 
 |       /      1  \ | 
 |       |1 + -----| | 
 |       |      ___| | 
 | 1     \    \/ x / | 
-|---- + ------------| 
 | 3/2           ___ | 
 \x      x + 2*\/ x  / 
-----------------------
        _____________  
       /         ___   
   4*\/  x + 2*\/ x

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

Simplify

The third derivative [src]

  /                   3                      \
  |        /      1  \               1       |
  |        |1 + -----|         1 + -----     |
  |        |      ___|               ___     |
  | 1      \    \/ x /             \/ x      |
3*|---- + -------------- + ------------------|
  | 5/2                2    3/2 /        ___\|
  |x      /        ___\    x   *\x + 2*\/ x /|
  \       \x + 2*\/ x /                      /
----------------------------------------------
                   _____________              
                  /         ___               
              8*\/  x + 2*\/ x

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

Simplify

The graph

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Derivative of y=sqrt(x+2*(sqrt(x)))

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The solution