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Identical expressions
y=sqrt(x)/(e^x+ one)
y equally square root of (x) divide by (e to the power of x plus 1)
y equally square root of (x) divide by (e to the power of x plus one)
y=√(x)/(e^x+1)
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y=sqrtx/e^x+1
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Similar expressions
y=sqrt(x)/(e^x-1)

The derivative of the function/ y=sqrt(x)/(e^x+1)

Derivative of y=sqrt(x)/(e^x+1)

The solution

You have entered [src]

  ___ 
\/ x  
------
 x    
e  + 1

$$\frac{\sqrt{x}}{e^{x} + 1}$$

  /  ___ \
d |\/ x  |
--|------|
dx| x    |
  \e  + 1/

$$\frac{d}{d x} \frac{\sqrt{x}}{e^{x} + 1}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sqrt{x}$ and $g{\left(x \right)} = e^{x} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $e^{x} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of $e^{x}$ is itself.
  The result is: $e^{x}$
Now plug in to the quotient rule:

$\frac{- \sqrt{x} e^{x} + \frac{e^{x} + 1}{2 \sqrt{x}}}{\left(e^{x} + 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                 /        x         2*x \                       
                             ___ |     6*e       6*e    |  x     /        x \   
                           \/ x *|1 - ------ + ---------|*e      |     2*e  |  x
                                 |         x           2|      3*|1 - ------|*e 
                  x              |    1 + e    /     x\ |        |         x|   
  3            3*e               \             \1 + e / /        \    1 + e /   
------ + --------------- - --------------------------------- - -----------------
   5/2      3/2 /     x\                      x                     ___ /     x\
8*x      4*x   *\1 + e /                 1 + e                  2*\/ x *\1 + e /
--------------------------------------------------------------------------------
                                          x                                     
                                     1 + e

$$\frac{- \frac{\sqrt{x} \left(1 - \frac{6 e^{x}}{e^{x} + 1} + \frac{6 e^{2 x}}{\left(e^{x} + 1\right)^{2}}\right) e^{x}}{e^{x} + 1} - \frac{3 \cdot \left(1 - \frac{2 e^{x}}{e^{x} + 1}\right) e^{x}}{2 \sqrt{x} \left(e^{x} + 1\right)} + \frac{3 e^{x}}{4 x^{\frac{3}{2}} \left(e^{x} + 1\right)} + \frac{3}{8 x^{\frac{5}{2}}}}{e^{x} + 1}$$

Simplify

The graph

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Derivative of y=sqrt(x)/(e^x+1)

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Piecewise:

The solution