Derivative of x^arctgx

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 acot(x)
x

$$x^{\operatorname{acot}{\left(x \right)}}$$

x^acot(x)

Detail solution

Don't know the steps in finding this derivative.

But the derivative is

$\left(\log{\left(\operatorname{acot}{\left(x \right)} \right)} + 1\right) \operatorname{acot}^{\operatorname{acot}{\left(x \right)}}{\left(x \right)}$

The answer is:

$\left(\log{\left(\operatorname{acot}{\left(x \right)} \right)} + 1\right) \operatorname{acot}^{\operatorname{acot}{\left(x \right)}}{\left(x \right)}$

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The first derivative [src]

 acot(x) /acot(x)   log(x)\
x       *|------- - ------|
         |   x           2|
         \          1 + x /

$$x^{\operatorname{acot}{\left(x \right)}} \left(- \frac{\log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{acot}{\left(x \right)}}{x}\right)$$

Simplify

The second derivative [src]

         /                  2                                    \
 acot(x) |/log(x)   acot(x)\    acot(x)       2        2*x*log(x)|
x       *||------ - -------|  - ------- - ---------- + ----------|
         ||     2      x   |        2       /     2\           2 |
         |\1 + x           /       x      x*\1 + x /   /     2\  |
         \                                             \1 + x /  /

$$x^{\operatorname{acot}{\left(x \right)}} \left(\frac{2 x \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \left(\frac{\log{\left(x \right)}}{x^{2} + 1} - \frac{\operatorname{acot}{\left(x \right)}}{x}\right)^{2} - \frac{2}{x \left(x^{2} + 1\right)} - \frac{\operatorname{acot}{\left(x \right)}}{x^{2}}\right)$$

Simplify

The third derivative [src]

         /                    3                                                                                                                   2       \
 acot(x) |  /log(x)   acot(x)\        6       2*acot(x)    2*log(x)        3          /log(x)   acot(x)\ /acot(x)       2        2*x*log(x)\   8*x *log(x)|
x       *|- |------ - -------|  + --------- + --------- + --------- + ----------- + 3*|------ - -------|*|------- + ---------- - ----------| - -----------|
         |  |     2      x   |            2        3              2    2 /     2\     |     2      x   | |    2       /     2\           2 |            3 |
         |  \1 + x           /    /     2\        x       /     2\    x *\1 + x /     \1 + x           / |   x      x*\1 + x /   /     2\  |    /     2\  |
         \                        \1 + x /                \1 + x /                                       \                       \1 + x /  /    \1 + x /  /

$$x^{\operatorname{acot}{\left(x \right)}} \left(- \frac{8 x^{2} \log{\left(x \right)}}{\left(x^{2} + 1\right)^{3}} - \left(\frac{\log{\left(x \right)}}{x^{2} + 1} - \frac{\operatorname{acot}{\left(x \right)}}{x}\right)^{3} + 3 \left(\frac{\log{\left(x \right)}}{x^{2} + 1} - \frac{\operatorname{acot}{\left(x \right)}}{x}\right) \left(- \frac{2 x \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{2}{x \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(x \right)}}{x^{2}}\right) + \frac{2 \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{6}{\left(x^{2} + 1\right)^{2}} + \frac{3}{x^{2} \left(x^{2} + 1\right)} + \frac{2 \operatorname{acot}{\left(x \right)}}{x^{3}}\right)$$

Simplify

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Identical expressions

Derivative of x^arctgx

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