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Derivative of:
Derivative of sin(x)^2
Derivative of y/2
Derivative of x/5+5/x
Derivative of x^3/(x^2-9)
Identical expressions
y=e^sqrt(x/lnx)
y equally e to the power of square root of (x divide by lnx)
y=e^√(x/lnx)
y=esqrt(x/lnx)
y=esqrtx/lnx
y=e^sqrtx/lnx
y=e^sqrt(x divide by lnx)

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

Derivative of y=e^sqrt(x/lnx)

The solution

You have entered [src]

     ________
    /   x    
   /  ------ 
 \/   log(x) 
E

$$e^{\sqrt{\frac{x}{\log{\left(x \right)}}}}$$

E^(sqrt(x/log(x)))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                         ________       
                                        /   x           
    ________                           /  ------        
   /   x     /   1           1    \  \/   log(x)        
  /  ------ *|-------- - ---------|*e            *log(x)
\/   log(x)  |2*log(x)        2   |                     
             \           2*log (x)/                     
--------------------------------------------------------
                           x

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

Simplify

The second derivative [src]

/            2       ________             2         ________                      ________                      ________             \      ________
|/      1   \       /   x     /      1   \         /   x     /      1   \        /   x     /      2   \        /   x     /      1   \|     /   x    
||1 - ------|      /  ------ *|1 - ------|    2*  /  ------ *|1 - ------|   2*  /  ------ *|1 - ------|   2*  /  ------ *|1 - ------||    /  ------ 
|\    log(x)/    \/   log(x)  \    log(x)/      \/   log(x)  \    log(x)/     \/   log(x)  \    log(x)/     \/   log(x)  \    log(x)/|  \/   log(x) 
|------------- + -------------------------- - --------------------------- - --------------------------- + ---------------------------|*e            
\    log(x)                  x                             x                          x*log(x)                      x*log(x)         /              
----------------------------------------------------------------------------------------------------------------------------------------------------
                                                                        4*x

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

Simplify

The third derivative [src]

/                                                                                                                                                                                                                                   ________                                                                                                                                                  \              
|            2               2               3       ________                      ________             2                                     ________             3       ________             3       ________                   /   x     /       6   \       ________                      ________                      ________             2         ________                          |      ________
|/      1   \    /      1   \    /      1   \       /   x     /      1   \        /   x     /      1   \      /      1   \ /      2   \      /   x     /      1   \       /   x     /      1   \       /   x     /      2   \     /  ------ *|1 - -------|      /   x     /      2   \        /   x     /      1   \        /   x     /      1   \         /   x     /      1   \ /      2   \|     /   x    
||1 - ------|    |1 - ------|    |1 - ------|      /  ------ *|1 - ------|   3*  /  ------ *|1 - ------|    3*|1 - ------|*|1 - ------|     /  ------ *|1 - ------|      /  ------ *|1 - ------|      /  ------ *|1 - ------|   \/   log(x)  |       2   |     /  ------ *|1 - ------|   3*  /  ------ *|1 - ------|   3*  /  ------ *|1 - ------|    3*  /  ------ *|1 - ------|*|1 - ------||    /  ------ 
|\    log(x)/    \    log(x)/    \    log(x)/    \/   log(x)  \    log(x)/     \/   log(x)  \    log(x)/      \    log(x)/ \    log(x)/   \/   log(x)  \    log(x)/    \/   log(x)  \    log(x)/    \/   log(x)  \    log(x)/                \    log (x)/   \/   log(x)  \    log(x)/     \/   log(x)  \    log(x)/     \/   log(x)  \    log(x)/      \/   log(x)  \    log(x)/ \    log(x)/|  \/   log(x) 
|------------- - ------------- + ------------- + ------------------------- - ---------------------------- - --------------------------- + -------------------------- + -------------------------- + ------------------------- + -------------------------- - ------------------------- - --------------------------- + ---------------------------- - ----------------------------------------|*e            
|       2           2*log(x)        8*log(x)                 x                           4*x                              2                          8*x                        8*log(x)                     x*log(x)                   2*x*log(x)                        2                       2*x*log(x)                    4*x*log(x)                           4*x*log(x)               |              
\  2*log (x)                                                                                                         4*log (x)                                                                                                                                       x*log (x)                                                                                                                /              
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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                                                                                                                                                                                                      x

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

Simplify

The graph

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How to use it?

Identical expressions

Derivative of y=e^sqrt(x/lnx)

The graph:

Piecewise:

The solution