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

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

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

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           2   
        cot (x)
(tan(x))
tan⁡cot⁡2(x)(x)\tan^{\cot^{2}{\left(x \right)}}{\left(x \right)} 
tan(x)^(cot(x)^2)
Detail solution

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

    But the derivative is

    (log⁡(cot⁡2(x))+1)∣cot⁡(x)∣2cot⁡2(x)\left(\log{\left(\cot^{2}{\left(x \right)} \right)} + 1\right) \left|{\cot{\left(x \right)}}\right|^{2 \cot^{2}{\left(x \right)}}

The answer is:

(log⁡(cot⁡2(x))+1)∣cot⁡(x)∣2cot⁡2(x)\left(\log{\left(\cot^{2}{\left(x \right)} \right)} + 1\right) \left|{\cot{\left(x \right)}}\right|^{2 \cot^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
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)                                             /
((tan⁡2(x)+1)cot⁡2(x)tan⁡(x)+(−2cot⁡2(x)−2)log⁡(tan⁡(x))cot⁡(x))tan⁡cot⁡2(x)(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)}
Simplify
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)                                                                                     /
(((tan⁡2(x)+1)cot⁡(x)tan⁡(x)−2(cot⁡2(x)+1)log⁡(tan⁡(x)))2cot⁡2(x)−(tan⁡2(x)+1)2cot⁡2(x)tan⁡2(x)−4(tan⁡2(x)+1)(cot⁡2(x)+1)cot⁡(x)tan⁡(x)+2(tan⁡2(x)+1)cot⁡2(x)+2(cot⁡2(x)+1)2log⁡(tan⁡(x))+4(cot⁡2(x)+1)log⁡(tan⁡(x))cot⁡2(x))tan⁡cot⁡2(x)(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)}
Simplify
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)                                                        /
(((tan⁡2(x)+1)cot⁡(x)tan⁡(x)−2(cot⁡2(x)+1)log⁡(tan⁡(x)))3cot⁡3(x)+3((tan⁡2(x)+1)cot⁡(x)tan⁡(x)−2(cot⁡2(x)+1)log⁡(tan⁡(x)))(−(tan⁡2(x)+1)2cot⁡2(x)tan⁡2(x)−4(tan⁡2(x)+1)(cot⁡2(x)+1)cot⁡(x)tan⁡(x)+2(tan⁡2(x)+1)cot⁡2(x)+2(cot⁡2(x)+1)2log⁡(tan⁡(x))+4(cot⁡2(x)+1)log⁡(tan⁡(x))cot⁡2(x))cot⁡(x)+2(tan⁡2(x)+1)3cot⁡2(x)tan⁡3(x)+6(tan⁡2(x)+1)2(cot⁡2(x)+1)cot⁡(x)tan⁡2(x)−4(tan⁡2(x)+1)2cot⁡2(x)tan⁡(x)+6(tan⁡2(x)+1)(cot⁡2(x)+1)2tan⁡(x)−12(tan⁡2(x)+1)(cot⁡2(x)+1)cot⁡(x)+12(tan⁡2(x)+1)(cot⁡2(x)+1)cot⁡2(x)tan⁡(x)+4(tan⁡2(x)+1)tan⁡(x)cot⁡2(x)−16(cot⁡2(x)+1)2log⁡(tan⁡(x))cot⁡(x)−8(cot⁡2(x)+1)log⁡(tan⁡(x))cot⁡3(x))tan⁡cot⁡2(x)(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)}
Simplify
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)                                                        /
(((tan⁡2(x)+1)cot⁡(x)tan⁡(x)−2(cot⁡2(x)+1)log⁡(tan⁡(x)))3cot⁡3(x)+3((tan⁡2(x)+1)cot⁡(x)tan⁡(x)−2(cot⁡2(x)+1)log⁡(tan⁡(x)))(−(tan⁡2(x)+1)2cot⁡2(x)tan⁡2(x)−4(tan⁡2(x)+1)(cot⁡2(x)+1)cot⁡(x)tan⁡(x)+2(tan⁡2(x)+1)cot⁡2(x)+2(cot⁡2(x)+1)2log⁡(tan⁡(x))+4(cot⁡2(x)+1)log⁡(tan⁡(x))cot⁡2(x))cot⁡(x)+2(tan⁡2(x)+1)3cot⁡2(x)tan⁡3(x)+6(tan⁡2(x)+1)2(cot⁡2(x)+1)cot⁡(x)tan⁡2(x)−4(tan⁡2(x)+1)2cot⁡2(x)tan⁡(x)+6(tan⁡2(x)+1)(cot⁡2(x)+1)2tan⁡(x)−12(tan⁡2(x)+1)(cot⁡2(x)+1)cot⁡(x)+12(tan⁡2(x)+1)(cot⁡2(x)+1)cot⁡2(x)tan⁡(x)+4(tan⁡2(x)+1)tan⁡(x)cot⁡2(x)−16(cot⁡2(x)+1)2log⁡(tan⁡(x))cot⁡(x)−8(cot⁡2(x)+1)log⁡(tan⁡(x))cot⁡3(x))tan⁡cot⁡2(x)(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)}
Simplify
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