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- Derivative of a implicit defined function
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How to use it?
- Derivative of:
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Derivative of 3*x^2-1/x^3
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Derivative of (1+cosx)/(1-cosx)
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Derivative of (1/2)*x
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Derivative of 1-1/x
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Identical expressions
- y=(tg two x)^ctgx/2
- y equally (tg2x) to the power of ctgx divide by 2
- y equally (tg two x) to the power of ctgx divide by 2
- y=(tg2x)ctgx/2
- y=tg2xctgx/2
- y=tg2x^ctgx/2
- y=(tg2x)^ctgx divide by 2
Derivative of y=(tg2x)^ctgx/2
The solution
You have entered
[src]
cot(x)
tan (2*x)
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2
$$\frac{\tan^{\cot{\left(x \right)}}{\left(2 x \right)}}{2}$$
tan(2*x)^cot(x)/2
Detail solution
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The derivative of a constant times a function is the constant times the derivative of the function.
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Don't know the steps in finding this derivative.
But the derivative is
So, the result is:
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The answer is:
The first derivative
[src]
/ / 2 \ \
cot(x) |/ 2 \ \2 + 2*tan (2*x)/*cot(x)|
tan (2*x)*|\-1 - cot (x)/*log(tan(2*x)) + ------------------------|
\ tan(2*x) /
------------------------------------------------------------------------
2
$$\frac{\left(\frac{\left(2 \tan^{2}{\left(2 x \right)} + 2\right) \cot{\left(x \right)}}{\tan{\left(2 x \right)}} + \left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(\tan{\left(2 x \right)} \right)}\right) \tan^{\cot{\left(x \right)}}{\left(2 x \right)}}{2}$$
The second derivative
[src]
/ 2 2 \
|/ / 2 \ \ / 2 \ / 2 \ / 2 \ |
cot(x) || / 2 \ 2*\1 + tan (2*x)/*cot(x)| / 2 \ 4*\1 + tan (2*x)/ *cot(x) 4*\1 + cot (x)/*\1 + tan (2*x)/ / 2 \ |
tan (2*x)*||- \1 + cot (x)/*log(tan(2*x)) + ------------------------| + 8*\1 + tan (2*x)/*cot(x) - ------------------------- - ------------------------------- + 2*\1 + cot (x)/*cot(x)*log(tan(2*x))|
|\ tan(2*x) / 2 tan(2*x) |
\ tan (2*x) /
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2
$$\frac{\left(\left(\frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(2 x \right)}} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(2 x \right)} \right)}\right)^{2} - \frac{4 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} \cot{\left(x \right)}}{\tan^{2}{\left(2 x \right)}} - \frac{4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\tan{\left(2 x \right)}} + 8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \cot{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(2 x \right)} \right)} \cot{\left(x \right)}\right) \tan^{\cot{\left(x \right)}}{\left(2 x \right)}}{2}$$
The third derivative
[src]
/ 3 / 2 \ 2 2 3 \
|/ / 2 \ \ / / 2 \ \ | / 2 \ / 2 \ / 2 \| 2 / 2 \ / 2 \ / 2 \ / 2 \ / 2 \ / 2 \ |
cot(x) || / 2 \ 2*\1 + tan (2*x)/*cot(x)| / 2 \ / 2 \ | / 2 \ 2*\1 + tan (2*x)/*cot(x)| | / 2 \ / 2 \ 2*\1 + tan (2*x)/ *cot(x) 2*\1 + cot (x)/*\1 + tan (2*x)/| / 2 \ 32*\1 + tan (2*x)/ *cot(x) 2 / 2 \ 12*\1 + tan (2*x)/ *\1 + cot (x)/ 16*\1 + tan (2*x)/ *cot(x) / 2 \ 12*\1 + cot (x)/*\1 + tan (2*x)/*cot(x)|
tan (2*x)*||- \1 + cot (x)/*log(tan(2*x)) + ------------------------| - 24*\1 + cot (x)/*\1 + tan (2*x)/ - 6*|- \1 + cot (x)/*log(tan(2*x)) + ------------------------|*|- 4*\1 + tan (2*x)/*cot(x) - \1 + cot (x)/*cot(x)*log(tan(2*x)) + ------------------------- + -------------------------------| - 2*\1 + cot (x)/ *log(tan(2*x)) - -------------------------- - 4*cot (x)*\1 + cot (x)/*log(tan(2*x)) + --------------------------------- + -------------------------- + 32*\1 + tan (2*x)/*cot(x)*tan(2*x) + ---------------------------------------|
|\ tan(2*x) / \ tan(2*x) / | 2 tan(2*x) | tan(2*x) 2 3 tan(2*x) |
\ \ tan (2*x) / tan (2*x) tan (2*x) /
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2
$$\frac{\left(\left(\frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(2 x \right)}} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(2 x \right)} \right)}\right)^{3} - 6 \left(\frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(2 x \right)}} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(2 x \right)} \right)}\right) \left(\frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} \cot{\left(x \right)}}{\tan^{2}{\left(2 x \right)}} + \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\tan{\left(2 x \right)}} - 4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \cot{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(2 x \right)} \right)} \cot{\left(x \right)}\right) + \frac{16 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{3} \cot{\left(x \right)}}{\tan^{3}{\left(2 x \right)}} + \frac{12 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(2 x \right)}} - \frac{32 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} \cot{\left(x \right)}}{\tan{\left(2 x \right)}} - 24 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) + \frac{12 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(2 x \right)}} + 32 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} \cot{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(2 x \right)} \right)} - 4 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(2 x \right)} \right)} \cot^{2}{\left(x \right)}\right) \tan^{\cot{\left(x \right)}}{\left(2 x \right)}}{2}$$
The graph