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Derivative of (atan(3x))^(acot(x))

Function f() - derivative -N order at the point
v

The graph:

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Piecewise:

The solution

You have entered [src]
    acot(x)     
atan       (3*x)
$$\operatorname{atan}^{\operatorname{acot}{\left(x \right)}}{\left(3 x \right)}$$
atan(3*x)^acot(x)
Detail solution
  1. Don't know the steps in finding this derivative.

    But the derivative is


The answer is:

The graph
The first derivative [src]
    acot(x)      /  log(atan(3*x))        3*acot(x)      \
atan       (3*x)*|- -------------- + --------------------|
                 |           2       /       2\          |
                 \      1 + x        \1 + 9*x /*atan(3*x)/
$$\left(\frac{3 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{x^{2} + 1}\right) \operatorname{atan}^{\operatorname{acot}{\left(x \right)}}{\left(3 x \right)}$$
The second derivative [src]
                 /                                         2                                                                                                      \
    acot(x)      |/  log(atan(3*x))        3*acot(x)      \          9*acot(x)                        6                 2*x*log(atan(3*x))        54*x*acot(x)    |
atan       (3*x)*||- -------------- + --------------------|  - ---------------------- - ----------------------------- + ------------------ - ---------------------|
                 ||           2       /       2\          |              2              /     2\ /       2\                         2                  2          |
                 |\      1 + x        \1 + 9*x /*atan(3*x)/    /       2\      2        \1 + x /*\1 + 9*x /*atan(3*x)       /     2\         /       2\           |
                 \                                             \1 + 9*x / *atan (3*x)                                       \1 + x /         \1 + 9*x / *atan(3*x)/
$$\left(- \frac{54 x \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{2} \operatorname{atan}{\left(3 x \right)}} + \frac{2 x \log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{\left(x^{2} + 1\right)^{2}} + \left(\frac{3 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{x^{2} + 1}\right)^{2} - \frac{9 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(3 x \right)}} - \frac{6}{\left(x^{2} + 1\right) \left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}}\right) \operatorname{atan}^{\operatorname{acot}{\left(x \right)}}{\left(3 x \right)}$$
The third derivative [src]
                 /                                         3                                                                                                                                                                                                       2                                                                                                                                                                                 2           \
    acot(x)      |/  log(atan(3*x))        3*acot(x)      \      /  log(atan(3*x))        3*acot(x)      \ /  2*x*log(atan(3*x))                 6                       9*acot(x)               54*x*acot(x)    \   2*log(atan(3*x))         54*acot(x)        8*x *log(atan(3*x))                  27                       54*acot(x)                      18*x                            162*x                    486*x*acot(x)           1944*x *acot(x)   |
atan       (3*x)*||- -------------- + --------------------|  - 3*|- -------------- + --------------------|*|- ------------------ + ----------------------------- + ---------------------- + ---------------------| + ---------------- - --------------------- - ------------------- + ------------------------------- + ---------------------- + ------------------------------ + ------------------------------ + ---------------------- + ---------------------|
                 ||           2       /       2\          |      |           2       /       2\          | |              2        /     2\ /       2\                       2                        2          |              2                 2                          3                           2                        3                      2                                           2                       3                        3          |
                 |\      1 + x        \1 + 9*x /*atan(3*x)/      \      1 + x        \1 + 9*x /*atan(3*x)/ |      /     2\         \1 + x /*\1 + 9*x /*atan(3*x)   /       2\      2        /       2\           |      /     2\        /       2\                   /     2\         /     2\ /       2\      2        /       2\      3        /     2\  /       2\             /     2\ /       2\              /       2\      2        /       2\           |
                 \                                                                                         \      \1 + x /                                         \1 + 9*x / *atan (3*x)   \1 + 9*x / *atan(3*x)/      \1 + x /        \1 + 9*x / *atan(3*x)        \1 + x /         \1 + x /*\1 + 9*x / *atan (3*x)   \1 + 9*x / *atan (3*x)   \1 + x / *\1 + 9*x /*atan(3*x)   \1 + x /*\1 + 9*x / *atan(3*x)   \1 + 9*x / *atan (3*x)   \1 + 9*x / *atan(3*x)/
$$\left(\frac{1944 x^{2} \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{3} \operatorname{atan}{\left(3 x \right)}} - \frac{8 x^{2} \log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{\left(x^{2} + 1\right)^{3}} + \frac{486 x \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{3} \operatorname{atan}^{2}{\left(3 x \right)}} + \frac{162 x}{\left(x^{2} + 1\right) \left(9 x^{2} + 1\right)^{2} \operatorname{atan}{\left(3 x \right)}} + \frac{18 x}{\left(x^{2} + 1\right)^{2} \left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} + \left(\frac{3 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{x^{2} + 1}\right)^{3} - 3 \left(\frac{3 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{x^{2} + 1}\right) \left(\frac{54 x \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{2} \operatorname{atan}{\left(3 x \right)}} - \frac{2 x \log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{9 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(3 x \right)}} + \frac{6}{\left(x^{2} + 1\right) \left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}}\right) - \frac{54 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{2} \operatorname{atan}{\left(3 x \right)}} + \frac{54 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{3} \operatorname{atan}^{3}{\left(3 x \right)}} + \frac{27}{\left(x^{2} + 1\right) \left(9 x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(3 x \right)}} + \frac{2 \log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{\left(x^{2} + 1\right)^{2}}\right) \operatorname{atan}^{\operatorname{acot}{\left(x \right)}}{\left(3 x \right)}$$