Mister Exam
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- Derivative of a implicit defined function
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How to use it?
- Derivative of:
- Derivative of e^x^x
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Derivative of ln(x+3)^3
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Derivative of cos(3*x)^(2)
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Derivative of -cos(2*x)
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Identical expressions
- (tan(x))^(cot(x)^ two)
- ( tangent of (x)) to the power of ( cotangent of (x) squared )
- ( tangent of (x)) to the power of ( cotangent of (x) to the power of two)
- (tan(x))(cot(x)2)
- tanxcotx2
- (tan(x))^(cot(x)²)
- (tan(x)) to the power of (cot(x) to the power of 2)
- tanx^cotx^2
Derivative of (tan(x))^(cot(x)^2)
The solution
You have entered
[src]
2
cot (x)
(tan(x))
$$\tan^{\cot^{2}{\left(x \right)}}{\left(x \right)}$$
tan(x)^(cot(x)^2)
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]
2 / 2 / 2 \ \
cot (x) |cot (x)*\1 + tan (x)/ / 2 \ |
(tan(x)) *|--------------------- + \-2 - 2*cot (x)/*cot(x)*log(tan(x))|
\ tan(x) /
$$\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)}}{\tan{\left(x \right)}} + \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \log{\left(\tan{\left(x \right)} \right)} \cot{\left(x \right)}\right) \tan^{\cot^{2}{\left(x \right)}}{\left(x \right)}$$
The second derivative
[src]
/ 2 2 \
2 |/ / 2 \ \ 2 / 2 \ 2 / 2 \ / 2 \ |
cot (x) || / 2 \ \1 + tan (x)/*cot(x)| 2 / 2 \ 2 / 2 \ \1 + tan (x)/ *cot (x) 2 / 2 \ 4*\1 + cot (x)/*\1 + tan (x)/*cot(x)|
(tan(x)) *||- 2*\1 + cot (x)/*log(tan(x)) + --------------------| *cot (x) + 2*\1 + cot (x)/ *log(tan(x)) + 2*cot (x)*\1 + tan (x)/ - ---------------------- + 4*cot (x)*\1 + cot (x)/*log(tan(x)) - ------------------------------------|
|\ tan(x) / 2 tan(x) |
\ tan (x) /
$$\left(\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}\right)^{2} \cot^{2}{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)} + 4 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)} \cot^{2}{\left(x \right)}\right) \tan^{\cot^{2}{\left(x \right)}}{\left(x \right)}$$
The third derivative
[src]
/ 3 2 3 / 2 \ 2 2 \
2 |/ / 2 \ \ 2 / 2 \ 2 / 2 \ 2 / / 2 \ \ | 2 / 2 \ 2 / 2 \ / 2 \ | / 2 \ / 2 \ / 2 \ / 2 \ 2 / 2 \ / 2 \|
cot (x) || / 2 \ \1 + tan (x)/*cot(x)| 3 / 2 \ / 2 \ / 2 \ 3 / 2 \ 4*\1 + tan (x)/ *cot (x) 2*\1 + tan (x)/ *cot (x) | / 2 \ \1 + tan (x)/*cot(x)| | / 2 \ 2 / 2 \ \1 + tan (x)/ *cot (x) 2 / 2 \ 4*\1 + cot (x)/*\1 + tan (x)/*cot(x)| 2 / 2 \ 6*\1 + cot (x)/ *\1 + tan (x)/ 6*\1 + tan (x)/ *\1 + cot (x)/*cot(x) 12*cot (x)*\1 + cot (x)/*\1 + tan (x)/|
(tan(x)) *||- 2*\1 + cot (x)/*log(tan(x)) + --------------------| *cot (x) - 16*\1 + cot (x)/ *cot(x)*log(tan(x)) - 12*\1 + cot (x)/*\1 + tan (x)/*cot(x) - 8*cot (x)*\1 + cot (x)/*log(tan(x)) - ------------------------ + ------------------------ + 3*|- 2*\1 + cot (x)/*log(tan(x)) + --------------------|*|2*\1 + cot (x)/ *log(tan(x)) + 2*cot (x)*\1 + tan (x)/ - ---------------------- + 4*cot (x)*\1 + cot (x)/*log(tan(x)) - ------------------------------------|*cot(x) + 4*cot (x)*\1 + tan (x)/*tan(x) + ------------------------------ + ------------------------------------- + --------------------------------------|
|\ tan(x) / tan(x) 3 \ tan(x) / | 2 tan(x) | tan(x) 2 tan(x) |
\ tan (x) \ tan (x) / tan (x) /
$$\left(\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}\right)^{3} \cot^{3}{\left(x \right)} + 3 \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}\right) \left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)} + 4 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)} \cot^{2}{\left(x \right)}\right) \cot{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \cot^{2}{\left(x \right)}}{\tan^{3}{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan^{2}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(x \right)}}{\tan{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\tan{\left(x \right)}} - 12 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{12 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)}}{\tan{\left(x \right)}} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot^{2}{\left(x \right)} - 16 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)} \cot{\left(x \right)} - 8 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)} \cot^{3}{\left(x \right)}\right) \tan^{\cot^{2}{\left(x \right)}}{\left(x \right)}$$
3-я производная
[src]
/ 3 2 3 / 2 \ 2 2 \
2 |/ / 2 \ \ 2 / 2 \ 2 / 2 \ 2 / / 2 \ \ | 2 / 2 \ 2 / 2 \ / 2 \ | / 2 \ / 2 \ / 2 \ / 2 \ 2 / 2 \ / 2 \|
cot (x) || / 2 \ \1 + tan (x)/*cot(x)| 3 / 2 \ / 2 \ / 2 \ 3 / 2 \ 4*\1 + tan (x)/ *cot (x) 2*\1 + tan (x)/ *cot (x) | / 2 \ \1 + tan (x)/*cot(x)| | / 2 \ 2 / 2 \ \1 + tan (x)/ *cot (x) 2 / 2 \ 4*\1 + cot (x)/*\1 + tan (x)/*cot(x)| 2 / 2 \ 6*\1 + cot (x)/ *\1 + tan (x)/ 6*\1 + tan (x)/ *\1 + cot (x)/*cot(x) 12*cot (x)*\1 + cot (x)/*\1 + tan (x)/|
(tan(x)) *||- 2*\1 + cot (x)/*log(tan(x)) + --------------------| *cot (x) - 16*\1 + cot (x)/ *cot(x)*log(tan(x)) - 12*\1 + cot (x)/*\1 + tan (x)/*cot(x) - 8*cot (x)*\1 + cot (x)/*log(tan(x)) - ------------------------ + ------------------------ + 3*|- 2*\1 + cot (x)/*log(tan(x)) + --------------------|*|2*\1 + cot (x)/ *log(tan(x)) + 2*cot (x)*\1 + tan (x)/ - ---------------------- + 4*cot (x)*\1 + cot (x)/*log(tan(x)) - ------------------------------------|*cot(x) + 4*cot (x)*\1 + tan (x)/*tan(x) + ------------------------------ + ------------------------------------- + --------------------------------------|
|\ tan(x) / tan(x) 3 \ tan(x) / | 2 tan(x) | tan(x) 2 tan(x) |
\ tan (x) \ tan (x) / tan (x) /
$$\left(\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}\right)^{3} \cot^{3}{\left(x \right)} + 3 \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}\right) \left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)} + 4 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)} \cot^{2}{\left(x \right)}\right) \cot{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \cot^{2}{\left(x \right)}}{\tan^{3}{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan^{2}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(x \right)}}{\tan{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\tan{\left(x \right)}} - 12 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{12 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)}}{\tan{\left(x \right)}} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot^{2}{\left(x \right)} - 16 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)} \cot{\left(x \right)} - 8 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)} \cot^{3}{\left(x \right)}\right) \tan^{\cot^{2}{\left(x \right)}}{\left(x \right)}$$