Mister Exam
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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 1/x^5
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Derivative of x^7*e^x
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Derivative of x^5*cos(x)
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Derivative of x/3+3/x
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Identical expressions
- (сtg(x))^(one /(ln(x)))
- (сtg(x)) to the power of (1 divide by (ln(x)))
- (сtg(x)) to the power of (one divide by (ln(x)))
- (сtg(x))(1/(ln(x)))
- сtgx1/lnx
- сtgx^1/lnx
- (сtg(x))^(1 divide by (ln(x)))
Derivative of (сtg(x))^(1/(ln(x)))
The solution
You have entered
[src]
1
------
log(x)
(cot(x))
$$\cot^{\frac{1}{\log{\left(x \right)}}}{\left(x \right)}$$
cot(x)^(1/log(x))
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
The answer is:
The first derivative
[src]
1
------ / 2 \
log(x) | -1 - cot (x) log(cot(x))|
(cot(x)) *|------------- - -----------|
|cot(x)*log(x) 2 |
\ x*log (x) /
$$\left(\frac{- \cot^{2}{\left(x \right)} - 1}{\log{\left(x \right)} \cot{\left(x \right)}} - \frac{\log{\left(\cot{\left(x \right)} \right)}}{x \log{\left(x \right)}^{2}}\right) \cot^{\frac{1}{\log{\left(x \right)}}}{\left(x \right)}$$
The second derivative
[src]
/ 2 \
| / 2 \ |
1 | |1 + cot (x) log(cot(x))| 2 |
------ | |----------- + -----------| / 2 \ / 2 \|
log(x) | 2 \ cot(x) x*log(x) / \1 + cot (x)/ log(cot(x)) 2*log(cot(x)) 2*\1 + cot (x)/|
(cot(x)) *|2 + 2*cot (x) + ---------------------------- - -------------- + ----------- + ------------- + ---------------|
| log(x) 2 2 2 2 x*cot(x)*log(x)|
\ cot (x) x *log(x) x *log (x) /
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log(x)
$$\frac{\left(\frac{\left(\frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}} + \frac{\log{\left(\cot{\left(x \right)} \right)}}{x \log{\left(x \right)}}\right)^{2}}{\log{\left(x \right)}} - \frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} + 2 \cot^{2}{\left(x \right)} + 2 + \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{x \log{\left(x \right)} \cot{\left(x \right)}} + \frac{\log{\left(\cot{\left(x \right)} \right)}}{x^{2} \log{\left(x \right)}} + \frac{2 \log{\left(\cot{\left(x \right)} \right)}}{x^{2} \log{\left(x \right)}^{2}}\right) \cot^{\frac{1}{\log{\left(x \right)}}}{\left(x \right)}}{\log{\left(x \right)}}$$
The third derivative
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/ / 2 \ \
| 3 / 2 \ | / 2 \ / 2 \| |
|/ 2 \ |1 + cot (x) log(cot(x))| | 2 \1 + cot (x)/ log(cot(x)) 2*log(cot(x)) 2*\1 + cot (x)/| |
1 ||1 + cot (x) log(cot(x))| 2 3 3*|----------- + -----------|*|2 + 2*cot (x) - -------------- + ----------- + ------------- + ---------------| 2 |
------ ||----------- + -----------| / 2 \ / 2 \ \ cot(x) x*log(x) / | 2 2 2 2 x*cot(x)*log(x)| / 2 \ / 2 \ / 2 \ / 2 \ |
log(x) |\ cot(x) x*log(x) / 4*\1 + cot (x)/ 2*\1 + cot (x)/ / 2 \ 2*log(cot(x)) \ cot (x) x *log(x) x *log (x) / 6*\1 + cot (x)/ 6*log(cot(x)) 6*log(cot(x)) 3*\1 + cot (x)/ 3*\1 + cot (x)/ 6*\1 + cot (x)/ |
-(cot(x)) *|---------------------------- - ---------------- + ---------------- + 4*\1 + cot (x)/*cot(x) + ------------- + -------------------------------------------------------------------------------------------------------------- + --------------- + ------------- + ------------- - ---------------- + ---------------- + -----------------|
| 2 cot(x) 3 3 log(x) x*log(x) 3 3 3 2 2 2 2 2 |
\ log (x) cot (x) x *log(x) x *log (x) x *log (x) x*cot (x)*log(x) x *cot(x)*log(x) x *cot(x)*log (x)/
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log(x)
$$- \frac{\left(\frac{\left(\frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}} + \frac{\log{\left(\cot{\left(x \right)} \right)}}{x \log{\left(x \right)}}\right)^{3}}{\log{\left(x \right)}^{2}} + \frac{3 \left(\frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}} + \frac{\log{\left(\cot{\left(x \right)} \right)}}{x \log{\left(x \right)}}\right) \left(- \frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} + 2 \cot^{2}{\left(x \right)} + 2 + \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{x \log{\left(x \right)} \cot{\left(x \right)}} + \frac{\log{\left(\cot{\left(x \right)} \right)}}{x^{2} \log{\left(x \right)}} + \frac{2 \log{\left(\cot{\left(x \right)} \right)}}{x^{2} \log{\left(x \right)}^{2}}\right)}{\log{\left(x \right)}} + \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)^{3}}{\cot^{3}{\left(x \right)}} - \frac{4 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot{\left(x \right)}} + 4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{x \log{\left(x \right)} \cot^{2}{\left(x \right)}} + \frac{6 \left(\cot^{2}{\left(x \right)} + 1\right)}{x \log{\left(x \right)}} + \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)} \cot{\left(x \right)}} + \frac{6 \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}^{2} \cot{\left(x \right)}} + \frac{2 \log{\left(\cot{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}} + \frac{6 \log{\left(\cot{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}^{2}} + \frac{6 \log{\left(\cot{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}^{3}}\right) \cot^{\frac{1}{\log{\left(x \right)}}}{\left(x \right)}}{\log{\left(x \right)}}$$
The graph