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- Derivative of a implicit defined function
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- Partial derivative of the function
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How to use it?
- Derivative of:
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Derivative of 50/x
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Derivative of e^sin^2x
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Derivative of cos(x)-tan(x)
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Derivative of arsinx
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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)
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
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Let .
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The derivative of is itself.
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Then, apply the chain rule. Multiply by :
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Let .
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Apply the power rule: goes to
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Then, apply the chain rule. Multiply by :
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Apply the quotient rule, which is:
and .
To find :
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Apply the power rule: goes to
To find :
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The derivative of is .
Now plug in to the quotient rule:
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The result of the chain rule is:
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The result of the chain rule is:
The answer is:
The first derivative
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________
/ x
________ / ------
/ x / 1 1 \ \/ log(x)
/ ------ *|-------- - ---------|*e *log(x)
\/ log(x) |2*log(x) 2 |
\ 2*log (x)/
--------------------------------------------------------
x
$$\frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(\frac{1}{2 \log{\left(x \right)}} - \frac{1}{2 \log{\left(x \right)}^{2}}\right) e^{\sqrt{\frac{x}{\log{\left(x \right)}}}} \log{\left(x \right)}}{x}$$
The second derivative
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/ 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
$$\frac{\left(\frac{\left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{\log{\left(x \right)}} - \frac{2 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{2}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}} + \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{x} - \frac{2 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{x} + \frac{2 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}}\right) e^{\sqrt{\frac{x}{\log{\left(x \right)}}}}}{4 x}$$
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) /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
x
$$\frac{\left(\frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)^{3}}{8 \log{\left(x \right)}} - \frac{3 \left(1 - \frac{2}{\log{\left(x \right)}}\right) \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{4 \log{\left(x \right)}^{2}} + \frac{\left(1 - \frac{1}{\log{\left(x \right)}}\right)^{3}}{8 \log{\left(x \right)}} - \frac{\left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{2 \log{\left(x \right)}} + \frac{\left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{2 \log{\left(x \right)}^{2}} + \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{6}{\log{\left(x \right)}^{2}}\right)}{2 x \log{\left(x \right)}} - \frac{3 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{2}{\log{\left(x \right)}}\right) \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{4 x \log{\left(x \right)}} + \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{2}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}} - \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{2}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}^{2}} + \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)^{3}}{8 x} - \frac{3 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{4 x} + \frac{3 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{4 x \log{\left(x \right)}} + \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{x} - \frac{3 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{2 x \log{\left(x \right)}}\right) e^{\sqrt{\frac{x}{\log{\left(x \right)}}}}}{x^{2}}$$
The graph