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√(x+√x)
The derivative of the function/ √(x+√x)

Derivative of √(x+√x)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   ___________
  /       ___ 
\/  x + \/ x
x+x\sqrt{\sqrt{x} + x} 
  /   ___________\
d |  /       ___ |
--\\/  x + \/ x  /
dx
ddxx+x\frac{d}{d x} \sqrt{\sqrt{x} + x}
Transform
Detail solution

  1. Let u=x+xu = \sqrt{x} + x.

  2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

  3. Then, apply the chain rule. Multiply by ddx(x+x)\frac{d}{d x} \left(\sqrt{x} + x\right):

    1. Differentiate x+x\sqrt{x} + x term by term:

      1. Apply the power rule: xx goes to 11

      2. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

      The result is: 1+12x1 + \frac{1}{2 \sqrt{x}}

    The result of the chain rule is:

    1+12x2x+x\frac{1 + \frac{1}{2 \sqrt{x}}}{2 \sqrt{\sqrt{x} + x}}

  4. Now simplify:

    2x+14xx+x\frac{2 \sqrt{x} + 1}{4 \sqrt{x} \sqrt{\sqrt{x} + x}}

The answer is:

2x+14xx+x\frac{2 \sqrt{x} + 1}{4 \sqrt{x} \sqrt{\sqrt{x} + x}}

The graph
02468-8-6-4-2-101005
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 1      1     
 - + -------  
 2       ___  
     4*\/ x   
--------------
   ___________
  /       ___ 
\/  x + \/ x
12+14xx+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
−(2+1x)2x+x+2x3216x+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
3((2+1x)3(x+x)2+2⋅(2+1x)x32(x+x)+4x52)64x+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
Derivative of √(x+√x)
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