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Similar expressions
(x^0,8*e^x)/(1-e^x)

The derivative of the function/ (x^0,8*e^x)/(1+e^x)

Derivative of (x^0,8*e^x)/(1+e^x)

The solution

You have entered [src]

 4/5  x
x   *e 
-------
      x
 1 + e

$$\frac{x^{\frac{4}{5}} e^{x}}{e^{x} + 1}$$

  / 4/5  x\
d |x   *e |
--|-------|
dx|      x|
  \ 1 + e /

$$\frac{d}{d x} \frac{x^{\frac{4}{5}} e^{x}}{e^{x} + 1}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{\frac{4}{5}} e^{x}$ and $g{\left(x \right)} = e^{x} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{\frac{4}{5}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{\frac{4}{5}}$ goes to $\frac{4}{5 \sqrt[5]{x}}$
  $g{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $e^{x}$ is itself.
  The result is: $x^{\frac{4}{5}} e^{x} + \frac{4 e^{x}}{5 \sqrt[5]{x}}$
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:

$\frac{- x^{\frac{4}{5}} e^{2 x} + \left(x^{\frac{4}{5}} e^{x} + \frac{4 e^{x}}{5 \sqrt[5]{x}}\right) \left(e^{x} + 1\right)}{\left(e^{x} + 1\right)^{2}}$
Now simplify:

$\frac{\left(5 x + 4 e^{x} + 4\right) e^{x}}{5 \sqrt[5]{x} \left(e^{2 x} + 2 e^{x} + 1\right)}$

The answer is:

$\frac{\left(5 x + 4 e^{x} + 4\right) e^{x}}{5 \sqrt[5]{x} \left(e^{2 x} + 2 e^{x} + 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 4/5  x    4/5  2*x            x      
x   *e    x   *e            4*e       
------- - --------- + ----------------
      x           2     5 ___ /     x\
 1 + e    /     x\    5*\/ x *\1 + e /
          \1 + e /

$$\frac{x^{\frac{4}{5}} e^{x}}{e^{x} + 1} - \frac{x^{\frac{4}{5}} e^{2 x}}{\left(e^{x} + 1\right)^{2}} + \frac{4 e^{x}}{5 \sqrt[5]{x} \left(e^{x} + 1\right)}$$

Simplify

The second derivative [src]

/                                                               /        x \   \   
|                                                           4/5 |     2*e  |  x|   
|                                                          x   *|1 - ------|*e |   
|                              4/5  x            x              |         x|   |   
| 4/5      4         8      2*x   *e          8*e               \    1 + e /   |  x
|x    - ------- + ------- - --------- - ---------------- - --------------------|*e 
|           6/5     5 ___          x      5 ___ /     x\               x       |   
\       25*x      5*\/ x      1 + e     5*\/ x *\1 + e /          1 + e        /   
-----------------------------------------------------------------------------------
                                            x                                      
                                       1 + e

$$\frac{\left(- \frac{x^{\frac{4}{5}} \cdot \left(1 - \frac{2 e^{x}}{e^{x} + 1}\right) e^{x}}{e^{x} + 1} + x^{\frac{4}{5}} - \frac{2 x^{\frac{4}{5}} e^{x}}{e^{x} + 1} + \frac{8}{5 \sqrt[5]{x}} - \frac{8 e^{x}}{5 \sqrt[5]{x} \left(e^{x} + 1\right)} - \frac{4}{25 x^{\frac{6}{5}}}\right) e^{x}}{e^{x} + 1}$$

Simplify

The third derivative [src]

/                                                                                              /        x         2*x \                                                 \   
|                                                                                          4/5 |     6*e       6*e    |  x          /        x \         /        x \   |   
|                                                                                         x   *|1 - ------ + ---------|*e       4/5 |     2*e  |  x      |     2*e  |  x|   
|                                                                                              |         x           2|      3*x   *|1 - ------|*e    12*|1 - ------|*e |   
|                                          4/5  x            x                  x              |    1 + e    /     x\ |             |         x|         |         x|   |   
| 4/5      12        12         24      3*x   *e         24*e               12*e               \             \1 + e / /             \    1 + e /         \    1 + e /   |  x
|x    - ------- + ------- + --------- - --------- - ---------------- + ---------------- - -------------------------------- - ---------------------- - ------------------|*e 
|           6/5     5 ___        11/5          x      5 ___ /     x\       6/5 /     x\                     x                             x              5 ___ /     x\ |   
\       25*x      5*\/ x    125*x         1 + e     5*\/ x *\1 + e /   25*x   *\1 + e /                1 + e                         1 + e             5*\/ x *\1 + e / /   
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                        x                                                                                   
                                                                                   1 + e

$$\frac{\left(- \frac{3 x^{\frac{4}{5}} \cdot \left(1 - \frac{2 e^{x}}{e^{x} + 1}\right) e^{x}}{e^{x} + 1} + x^{\frac{4}{5}} - \frac{x^{\frac{4}{5}} \cdot \left(1 - \frac{6 e^{x}}{e^{x} + 1} + \frac{6 e^{2 x}}{\left(e^{x} + 1\right)^{2}}\right) e^{x}}{e^{x} + 1} - \frac{3 x^{\frac{4}{5}} e^{x}}{e^{x} + 1} - \frac{12 \cdot \left(1 - \frac{2 e^{x}}{e^{x} + 1}\right) e^{x}}{5 \sqrt[5]{x} \left(e^{x} + 1\right)} + \frac{12}{5 \sqrt[5]{x}} - \frac{24 e^{x}}{5 \sqrt[5]{x} \left(e^{x} + 1\right)} - \frac{12}{25 x^{\frac{6}{5}}} + \frac{12 e^{x}}{25 x^{\frac{6}{5}} \left(e^{x} + 1\right)} + \frac{24}{125 x^{\frac{11}{5}}}\right) e^{x}}{e^{x} + 1}$$

Simplify

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Derivative of (x^0,8*e^x)/(1+e^x)

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The solution