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The derivative of the function/ ((arctgx)^coshx)

Derivative of ((arctgx)^coshx)

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

You have entered [src] 
    cosh(x)   
acot       (x)
acot⁡cosh⁡(x)(x)\operatorname{acot}^{\cosh{\left(x \right)}}{\left(x \right)} 
acot(x)^cosh(x)
Detail solution

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

    But the derivative is

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

The answer is:

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

The graph
02468-8-6-4-2-10102.5-2.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    cosh(x)    /                           cosh(x)     \
acot       (x)*|log(acot(x))*sinh(x) - ----------------|
               |                       /     2\        |
               \                       \1 + x /*acot(x)/
(log⁡(acot⁡(x))sinh⁡(x)−cosh⁡(x)(x2+1)acot⁡(x))acot⁡cosh⁡(x)(x)\left(\log{\left(\operatorname{acot}{\left(x \right)} \right)} \sinh{\left(x \right)} - \frac{\cosh{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)}}\right) \operatorname{acot}^{\cosh{\left(x \right)}}{\left(x \right)}
Simplify
The second derivative [src] 
               /                                         2                                                                                   \
    cosh(x)    |/                           cosh(x)     \                                cosh(x)            2*sinh(x)          2*x*cosh(x)   |
acot       (x)*||log(acot(x))*sinh(x) - ----------------|  + cosh(x)*log(acot(x)) - ------------------ - ---------------- + -----------------|
               ||                       /     2\        |                                   2            /     2\                   2        |
               |\                       \1 + x /*acot(x)/                           /     2\      2      \1 + x /*acot(x)   /     2\         |
               \                                                                    \1 + x / *acot (x)                      \1 + x / *acot(x)/
(2xcosh⁡(x)(x2+1)2acot⁡(x)+(log⁡(acot⁡(x))sinh⁡(x)−cosh⁡(x)(x2+1)acot⁡(x))2+log⁡(acot⁡(x))cosh⁡(x)−2sinh⁡(x)(x2+1)acot⁡(x)−cosh⁡(x)(x2+1)2acot⁡2(x))acot⁡cosh⁡(x)(x)\left(\frac{2 x \cosh{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}{\left(x \right)}} + \left(\log{\left(\operatorname{acot}{\left(x \right)} \right)} \sinh{\left(x \right)} - \frac{\cosh{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)}}\right)^{2} + \log{\left(\operatorname{acot}{\left(x \right)} \right)} \cosh{\left(x \right)} - \frac{2 \sinh{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)}} - \frac{\cosh{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}^{2}{\left(x \right)}}\right) \operatorname{acot}^{\cosh{\left(x \right)}}{\left(x \right)}
Simplify
The third derivative [src] 
               /                                         3                                                                                                                                                                                                                                                  2                                                   \
    cosh(x)    |/                           cosh(x)     \                             /                           cosh(x)     \ /                            cosh(x)            2*sinh(x)          2*x*cosh(x)   \      3*cosh(x)           3*sinh(x)            2*cosh(x)            2*cosh(x)          8*x *cosh(x)        6*x*cosh(x)          6*x*sinh(x)   |
acot       (x)*||log(acot(x))*sinh(x) - ----------------|  + log(acot(x))*sinh(x) + 3*|log(acot(x))*sinh(x) - ----------------|*|cosh(x)*log(acot(x)) - ------------------ - ---------------- + -----------------| - ---------------- - ------------------ - ------------------ + ----------------- - ----------------- + ------------------ + -----------------|
               ||                       /     2\        |                             |                       /     2\        | |                               2            /     2\                   2        |   /     2\                   2                    3                    2                   3                   3                    2        |
               |\                       \1 + x /*acot(x)/                             \                       \1 + x /*acot(x)/ |                       /     2\      2      \1 + x /*acot(x)   /     2\         |   \1 + x /*acot(x)   /     2\      2      /     2\      3      /     2\            /     2\            /     2\      2      /     2\         |
               \                                                                                                                \                       \1 + x / *acot (x)                      \1 + x / *acot(x)/                      \1 + x / *acot (x)   \1 + x / *acot (x)   \1 + x / *acot(x)   \1 + x / *acot(x)   \1 + x / *acot (x)   \1 + x / *acot(x)/
(−8x2cosh⁡(x)(x2+1)3acot⁡(x)+6xsinh⁡(x)(x2+1)2acot⁡(x)+6xcosh⁡(x)(x2+1)3acot⁡2(x)+(log⁡(acot⁡(x))sinh⁡(x)−cosh⁡(x)(x2+1)acot⁡(x))3+3(log⁡(acot⁡(x))sinh⁡(x)−cosh⁡(x)(x2+1)acot⁡(x))(2xcosh⁡(x)(x2+1)2acot⁡(x)+log⁡(acot⁡(x))cosh⁡(x)−2sinh⁡(x)(x2+1)acot⁡(x)−cosh⁡(x)(x2+1)2acot⁡2(x))+log⁡(acot⁡(x))sinh⁡(x)−3cosh⁡(x)(x2+1)acot⁡(x)−3sinh⁡(x)(x2+1)2acot⁡2(x)+2cosh⁡(x)(x2+1)2acot⁡(x)−2cosh⁡(x)(x2+1)3acot⁡3(x))acot⁡cosh⁡(x)(x)\left(- \frac{8 x^{2} \cosh{\left(x \right)}}{\left(x^{2} + 1\right)^{3} \operatorname{acot}{\left(x \right)}} + \frac{6 x \sinh{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}{\left(x \right)}} + \frac{6 x \cosh{\left(x \right)}}{\left(x^{2} + 1\right)^{3} \operatorname{acot}^{2}{\left(x \right)}} + \left(\log{\left(\operatorname{acot}{\left(x \right)} \right)} \sinh{\left(x \right)} - \frac{\cosh{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)}}\right)^{3} + 3 \left(\log{\left(\operatorname{acot}{\left(x \right)} \right)} \sinh{\left(x \right)} - \frac{\cosh{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)}}\right) \left(\frac{2 x \cosh{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}{\left(x \right)}} + \log{\left(\operatorname{acot}{\left(x \right)} \right)} \cosh{\left(x \right)} - \frac{2 \sinh{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)}} - \frac{\cosh{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}^{2}{\left(x \right)}}\right) + \log{\left(\operatorname{acot}{\left(x \right)} \right)} \sinh{\left(x \right)} - \frac{3 \cosh{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)}} - \frac{3 \sinh{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}^{2}{\left(x \right)}} + \frac{2 \cosh{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}{\left(x \right)}} - \frac{2 \cosh{\left(x \right)}}{\left(x^{2} + 1\right)^{3} \operatorname{acot}^{3}{\left(x \right)}}\right) \operatorname{acot}^{\cosh{\left(x \right)}}{\left(x \right)}
Simplify
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