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Limit of the function:
exp(-sqrt(x))/sqrt(x)
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exp(-sqrt(x))/sqrt(x)
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exp(-sqrt(x))/sqrt(x)
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Similar expressions
exp(sqrt(x))/sqrt(x)

The derivative of the function/ exp(-sqrt(x))/sqrt(x)

Derivative of exp(-sqrt(x))/sqrt(x)

The solution

You have entered [src]

    ___
 -\/ x 
e      
-------
   ___ 
 \/ x

$$\frac{e^{- \sqrt{x}}}{\sqrt{x}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  $g{\left(x \right)} = e^{\sqrt{x}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \sqrt{x}$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x}$ :
    1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    The result of the chain rule is:
    
    $\frac{e^{\sqrt{x}}}{2 \sqrt{x}}$
  The result is: $\frac{e^{\sqrt{x}}}{2} + \frac{e^{\sqrt{x}}}{2 \sqrt{x}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

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The first derivative [src]

      ___          ___   
   -\/ x        -\/ x    
  e            e         
- ------- - -------------
      3/2       ___   ___
   2*x      2*\/ x *\/ x

$$- \frac{e^{- \sqrt{x}}}{2 \sqrt{x} \sqrt{x}} - \frac{e^{- \sqrt{x}}}{2 x^{\frac{3}{2}}}$$

Simplify

The second derivative [src]

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

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

Simplify

4-я производная [src]

/            1     6     15    15      / 1     3     3  \      /1    1  \\        
|            -- + ---- + -- + ----   4*|---- + -- + ----|   18*|- + ----||        
|             2    5/2    3    7/2     | 3/2    2    5/2|      |x    3/2||     ___
|60   105    x    x      x    x        \x      x    x   /      \    x   /|  -\/ x 
|-- + ---- + --------------------- + -------------------- + -------------|*e      
| 4    9/2             ___                    3/2                 5/2    |        
\x    x              \/ x                    x                   x       /        
----------------------------------------------------------------------------------
                                        16

$$\frac{\left(\frac{60}{x^{4}} + \frac{\frac{1}{x^{2}} + \frac{15}{x^{3}} + \frac{6}{x^{\frac{5}{2}}} + \frac{15}{x^{\frac{7}{2}}}}{\sqrt{x}} + \frac{4 \left(\frac{3}{x^{2}} + \frac{1}{x^{\frac{3}{2}}} + \frac{3}{x^{\frac{5}{2}}}\right)}{x^{\frac{3}{2}}} + \frac{18 \left(\frac{1}{x} + \frac{1}{x^{\frac{3}{2}}}\right)}{x^{\frac{5}{2}}} + \frac{105}{x^{\frac{9}{2}}}\right) e^{- \sqrt{x}}}{16}$$

Simplify

The third derivative [src]

 /             1     3     3       /1    1  \\         
 |            ---- + -- + ----   3*|- + ----||         
 |             3/2    2    5/2     |x    3/2||     ___ 
 |9     15    x      x    x        \    x   /|  -\/ x  
-|-- + ---- + ---------------- + ------------|*e       
 | 3    7/2          ___              3/2    |         
 \x    x           \/ x              x       /         
-------------------------------------------------------
                           8

$$- \frac{\left(\frac{9}{x^{3}} + \frac{\frac{3}{x^{2}} + \frac{1}{x^{\frac{3}{2}}} + \frac{3}{x^{\frac{5}{2}}}}{\sqrt{x}} + \frac{3 \left(\frac{1}{x} + \frac{1}{x^{\frac{3}{2}}}\right)}{x^{\frac{3}{2}}} + \frac{15}{x^{\frac{7}{2}}}\right) e^{- \sqrt{x}}}{8}$$

Simplify

The graph

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Derivative of exp(-sqrt(x))/sqrt(x)

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Piecewise:

The solution