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The derivative of the function/ (cot(x)/2)^tan(2*x)

Derivative of (cot(x)/2)^tan(2*x)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
        tan(2*x)
/cot(x)\        
|------|        
\  2   /
(cot⁡(x)2)tan⁡(2x)\left(\frac{\cot{\left(x \right)}}{2}\right)^{\tan{\left(2 x \right)}} 
(cot(x)/2)^tan(2*x)
Detail solution

  1. Don't know the steps in finding this derivative.

    But the derivative is

    (log⁡(tan⁡(2x))+1)tan⁡tan⁡(2x)(2x)\left(\log{\left(\tan{\left(2 x \right)} \right)} + 1\right) \tan^{\tan{\left(2 x \right)}}{\left(2 x \right)}

The answer is:

(log⁡(tan⁡(2x))+1)tan⁡tan⁡(2x)(2x)\left(\log{\left(\tan{\left(2 x \right)} \right)} + 1\right) \tan^{\tan{\left(2 x \right)}}{\left(2 x \right)}

The graph
02468-8-6-4-2-1010-1000000000000000010000000000000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                 /                                  /         2   \         \
                 |                                  |  1   cot (x)|         |
        tan(2*x) |                                2*|- - - -------|*tan(2*x)|
/cot(x)\         |/         2     \    /cot(x)\     \  2      2   /         |
|------|        *|\2 + 2*tan (2*x)/*log|------| + --------------------------|
\  2   /         \                     \  2   /             cot(x)          /
(cot⁡(x)2)tan⁡(2x)((2tan⁡2(2x)+2)log⁡(cot⁡(x)2)+2(−cot⁡2(x)2−12)tan⁡(2x)cot⁡(x))\left(\frac{\cot{\left(x \right)}}{2}\right)^{\tan{\left(2 x \right)}} \left(\left(2 \tan^{2}{\left(2 x \right)} + 2\right) \log{\left(\frac{\cot{\left(x \right)}}{2} \right)} + \frac{2 \left(- \frac{\cot^{2}{\left(x \right)}}{2} - \frac{1}{2}\right) \tan{\left(2 x \right)}}{\cot{\left(x \right)}}\right)
Simplify
The second derivative [src] 
                 /                                                        2                                           2                                                                                    \
        tan(2*x) |/                                /       2   \         \                               /       2   \               /       2   \ /       2     \                                         |
/cot(x)\         ||  /       2     \    /cot(x)\   \1 + cot (x)/*tan(2*x)|      /       2   \            \1 + cot (x)/ *tan(2*x)   4*\1 + cot (x)/*\1 + tan (2*x)/     /       2     \    /cot(x)\         |
|------|        *||2*\1 + tan (2*x)/*log|------| - ----------------------|  + 2*\1 + cot (x)/*tan(2*x) - ----------------------- - ------------------------------- + 8*\1 + tan (2*x)/*log|------|*tan(2*x)|
\  2   /         |\                     \  2   /           cot(x)        /                                          2                           cot(x)                                    \  2   /         |
                 \                                                                                               cot (x)                                                                                   /
(cot⁡(x)2)tan⁡(2x)((2(tan⁡2(2x)+1)log⁡(cot⁡(x)2)−(cot⁡2(x)+1)tan⁡(2x)cot⁡(x))2−4(tan⁡2(2x)+1)(cot⁡2(x)+1)cot⁡(x)+8(tan⁡2(2x)+1)log⁡(cot⁡(x)2)tan⁡(2x)−(cot⁡2(x)+1)2tan⁡(2x)cot⁡2(x)+2(cot⁡2(x)+1)tan⁡(2x))\left(\frac{\cot{\left(x \right)}}{2}\right)^{\tan{\left(2 x \right)}} \left(\left(2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \log{\left(\frac{\cot{\left(x \right)}}{2} \right)} - \frac{\left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(2 x \right)}}{\cot{\left(x \right)}}\right)^{2} - \frac{4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}} + 8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \log{\left(\frac{\cot{\left(x \right)}}{2} \right)} \tan{\left(2 x \right)} - \frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(2 x \right)}}{\cot^{2}{\left(x \right)}} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(2 x \right)}\right)
Simplify
The third derivative [src] 
                 /                                                        3                                                              /                                          2                                                                                    \                                                                                       2                                                                    3                           2                                                                                                \
        tan(2*x) |/                                /       2   \         \      /                                /       2   \         \ |                             /       2   \                                                        /       2   \ /       2     \|                                                        2                 /       2   \  /       2     \                                       /       2   \               /       2   \                                                           /       2   \ /       2     \         |
/cot(x)\         ||  /       2     \    /cot(x)\   \1 + cot (x)/*tan(2*x)|      |  /       2     \    /cot(x)\   \1 + cot (x)/*tan(2*x)| |    /       2   \            \1 + cot (x)/ *tan(2*x)     /       2     \    /cot(x)\            4*\1 + cot (x)/*\1 + tan (2*x)/|      /       2   \ /       2     \      /       2     \     /cot(x)\   6*\1 + cot (x)/ *\1 + tan (2*x)/     /       2   \                   2*\1 + cot (x)/ *tan(2*x)   4*\1 + cot (x)/ *tan(2*x)         2      /       2     \    /cot(x)\   24*\1 + cot (x)/*\1 + tan (2*x)/*tan(2*x)|
|------|        *||2*\1 + tan (2*x)/*log|------| - ----------------------|  - 3*|2*\1 + tan (2*x)/*log|------| - ----------------------|*|- 2*\1 + cot (x)/*tan(2*x) + ----------------------- - 8*\1 + tan (2*x)/*log|------|*tan(2*x) + -------------------------------| + 12*\1 + cot (x)/*\1 + tan (2*x)/ + 16*\1 + tan (2*x)/ *log|------| - -------------------------------- - 4*\1 + cot (x)/*cot(x)*tan(2*x) - ------------------------- + ------------------------- + 32*tan (2*x)*\1 + tan (2*x)/*log|------| - -----------------------------------------|
\  2   /         |\                     \  2   /           cot(x)        /      \                     \  2   /           cot(x)        / |                                        2                                   \  2   /                         cot(x)            |                                                             \  2   /                  2                                                                 3                         cot(x)                                            \  2   /                     cot(x)                 |
                 \                                                                                                                       \                                     cot (x)                                                                                   /                                                                                    cot (x)                                                           cot (x)                                                                                                                            /
(cot⁡(x)2)tan⁡(2x)((2(tan⁡2(2x)+1)log⁡(cot⁡(x)2)−(cot⁡2(x)+1)tan⁡(2x)cot⁡(x))3−3(2(tan⁡2(2x)+1)log⁡(cot⁡(x)2)−(cot⁡2(x)+1)tan⁡(2x)cot⁡(x))(4(tan⁡2(2x)+1)(cot⁡2(x)+1)cot⁡(x)−8(tan⁡2(2x)+1)log⁡(cot⁡(x)2)tan⁡(2x)+(cot⁡2(x)+1)2tan⁡(2x)cot⁡2(x)−2(cot⁡2(x)+1)tan⁡(2x))+16(tan⁡2(2x)+1)2log⁡(cot⁡(x)2)−6(tan⁡2(2x)+1)(cot⁡2(x)+1)2cot⁡2(x)−24(tan⁡2(2x)+1)(cot⁡2(x)+1)tan⁡(2x)cot⁡(x)+12(tan⁡2(2x)+1)(cot⁡2(x)+1)+32(tan⁡2(2x)+1)log⁡(cot⁡(x)2)tan⁡2(2x)−2(cot⁡2(x)+1)3tan⁡(2x)cot⁡3(x)+4(cot⁡2(x)+1)2tan⁡(2x)cot⁡(x)−4(cot⁡2(x)+1)tan⁡(2x)cot⁡(x))\left(\frac{\cot{\left(x \right)}}{2}\right)^{\tan{\left(2 x \right)}} \left(\left(2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \log{\left(\frac{\cot{\left(x \right)}}{2} \right)} - \frac{\left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(2 x \right)}}{\cot{\left(x \right)}}\right)^{3} - 3 \left(2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \log{\left(\frac{\cot{\left(x \right)}}{2} \right)} - \frac{\left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(2 x \right)}}{\cot{\left(x \right)}}\right) \left(\frac{4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}} - 8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \log{\left(\frac{\cot{\left(x \right)}}{2} \right)} \tan{\left(2 x \right)} + \frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(2 x \right)}}{\cot^{2}{\left(x \right)}} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(2 x \right)}\right) + 16 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} \log{\left(\frac{\cot{\left(x \right)}}{2} \right)} - \frac{6 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} - \frac{24 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(2 x \right)}}{\cot{\left(x \right)}} + 12 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) + 32 \left(\tan^{2}{\left(2 x \right)} + 1\right) \log{\left(\frac{\cot{\left(x \right)}}{2} \right)} \tan^{2}{\left(2 x \right)} - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)^{3} \tan{\left(2 x \right)}}{\cot^{3}{\left(x \right)}} + \frac{4 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(2 x \right)}}{\cot{\left(x \right)}} - 4 \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(2 x \right)} \cot{\left(x \right)}\right)
Simplify
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