Derivative of sh3x^arctg(x+2)

The solution

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    atan(x + 2)     
sinh           (3*x)

$$\sinh^{\operatorname{atan}{\left(x + 2 \right)}}{\left(3 x \right)}$$

sinh(3*x)^atan(x + 2)

Detail solution

Don't know the steps in finding this derivative.

But the derivative is

$\left(\log{\left(\operatorname{atan}{\left(x + 2 \right)} \right)} + 1\right) \operatorname{atan}^{\operatorname{atan}{\left(x + 2 \right)}}{\left(x + 2 \right)}$
Now simplify:

$\left(\log{\left(\operatorname{atan}{\left(x + 2 \right)} \right)} + 1\right) \operatorname{atan}^{\operatorname{atan}{\left(x + 2 \right)}}{\left(x + 2 \right)}$

The answer is:

$\left(\log{\left(\operatorname{atan}{\left(x + 2 \right)} \right)} + 1\right) \operatorname{atan}^{\operatorname{atan}{\left(x + 2 \right)}}{\left(x + 2 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    atan(x + 2)      /log(sinh(3*x))   3*atan(x + 2)*cosh(3*x)\
sinh           (3*x)*|-------------- + -----------------------|
                     |            2           sinh(3*x)       |
                     \ 1 + (x + 2)                            /

$$\left(\frac{3 \cosh{\left(3 x \right)} \operatorname{atan}{\left(x + 2 \right)}}{\sinh{\left(3 x \right)}} + \frac{\log{\left(\sinh{\left(3 x \right)} \right)}}{\left(x + 2\right)^{2} + 1}\right) \sinh^{\operatorname{atan}{\left(x + 2 \right)}}{\left(3 x \right)}$$

Simplify

The second derivative [src]

                     /                                          2                         2                                                                       \
    atan(2 + x)      |/log(sinh(3*x))   3*atan(2 + x)*cosh(3*x)\                    9*cosh (3*x)*atan(2 + x)   2*(2 + x)*log(sinh(3*x))         6*cosh(3*x)       |
sinh           (3*x)*||-------------- + -----------------------|  + 9*atan(2 + x) - ------------------------ - ------------------------ + ------------------------|
                     ||            2           sinh(3*x)       |                               2                                 2        /           2\          |
                     |\ 1 + (2 + x)                            /                           sinh (3*x)              /           2\         \1 + (2 + x) /*sinh(3*x)|
                     \                                                                                             \1 + (2 + x) /                                 /

$$\left(- \frac{2 \left(x + 2\right) \log{\left(\sinh{\left(3 x \right)} \right)}}{\left(\left(x + 2\right)^{2} + 1\right)^{2}} + \left(\frac{3 \cosh{\left(3 x \right)} \operatorname{atan}{\left(x + 2 \right)}}{\sinh{\left(3 x \right)}} + \frac{\log{\left(\sinh{\left(3 x \right)} \right)}}{\left(x + 2\right)^{2} + 1}\right)^{2} + 9 \operatorname{atan}{\left(x + 2 \right)} - \frac{9 \cosh^{2}{\left(3 x \right)} \operatorname{atan}{\left(x + 2 \right)}}{\sinh^{2}{\left(3 x \right)}} + \frac{6 \cosh{\left(3 x \right)}}{\left(\left(x + 2\right)^{2} + 1\right) \sinh{\left(3 x \right)}}\right) \sinh^{\operatorname{atan}{\left(x + 2 \right)}}{\left(3 x \right)}$$

Simplify

The third derivative [src]

                     /                                          3                                                               /                                                                             2                 \                                                              2                       2                         3                                             \
    atan(2 + x)      |/log(sinh(3*x))   3*atan(2 + x)*cosh(3*x)\         27          /log(sinh(3*x))   3*atan(2 + x)*cosh(3*x)\ |                       6*cosh(3*x)          2*(2 + x)*log(sinh(3*x))   9*cosh (3*x)*atan(2 + x)|   2*log(sinh(3*x))   54*atan(2 + x)*cosh(3*x)         27*cosh (3*x)         8*(2 + x) *log(sinh(3*x))   54*cosh (3*x)*atan(2 + x)      18*(2 + x)*cosh(3*x)  |
sinh           (3*x)*||-------------- + -----------------------|  + ------------ - 3*|-------------- + -----------------------|*|-9*atan(2 + x) - ------------------------ + ------------------------ + ------------------------| - ---------------- - ------------------------ - ------------------------- + ------------------------- + ------------------------- - -------------------------|
                     ||            2           sinh(3*x)       |               2     |            2           sinh(3*x)       | |                 /           2\                               2                   2            |                 2           sinh(3*x)           /           2\     2                           3                    3                             2          |
                     |\ 1 + (2 + x)                            /    1 + (2 + x)      \ 1 + (2 + x)                            / |                 \1 + (2 + x) /*sinh(3*x)       /           2\                sinh (3*x)       |   /           2\                                \1 + (2 + x) /*sinh (3*x)        /           2\                 sinh (3*x)          /           2\           |
                     \                                                                                                          \                                                \1 + (2 + x) /                                 /   \1 + (2 + x) /                                                                 \1 + (2 + x) /                                     \1 + (2 + x) / *sinh(3*x)/

$$\left(\frac{8 \left(x + 2\right)^{2} \log{\left(\sinh{\left(3 x \right)} \right)}}{\left(\left(x + 2\right)^{2} + 1\right)^{3}} - \frac{18 \left(x + 2\right) \cosh{\left(3 x \right)}}{\left(\left(x + 2\right)^{2} + 1\right)^{2} \sinh{\left(3 x \right)}} + \left(\frac{3 \cosh{\left(3 x \right)} \operatorname{atan}{\left(x + 2 \right)}}{\sinh{\left(3 x \right)}} + \frac{\log{\left(\sinh{\left(3 x \right)} \right)}}{\left(x + 2\right)^{2} + 1}\right)^{3} - 3 \left(\frac{3 \cosh{\left(3 x \right)} \operatorname{atan}{\left(x + 2 \right)}}{\sinh{\left(3 x \right)}} + \frac{\log{\left(\sinh{\left(3 x \right)} \right)}}{\left(x + 2\right)^{2} + 1}\right) \left(\frac{2 \left(x + 2\right) \log{\left(\sinh{\left(3 x \right)} \right)}}{\left(\left(x + 2\right)^{2} + 1\right)^{2}} - 9 \operatorname{atan}{\left(x + 2 \right)} + \frac{9 \cosh^{2}{\left(3 x \right)} \operatorname{atan}{\left(x + 2 \right)}}{\sinh^{2}{\left(3 x \right)}} - \frac{6 \cosh{\left(3 x \right)}}{\left(\left(x + 2\right)^{2} + 1\right) \sinh{\left(3 x \right)}}\right) - \frac{54 \cosh{\left(3 x \right)} \operatorname{atan}{\left(x + 2 \right)}}{\sinh{\left(3 x \right)}} + \frac{54 \cosh^{3}{\left(3 x \right)} \operatorname{atan}{\left(x + 2 \right)}}{\sinh^{3}{\left(3 x \right)}} + \frac{27}{\left(x + 2\right)^{2} + 1} - \frac{27 \cosh^{2}{\left(3 x \right)}}{\left(\left(x + 2\right)^{2} + 1\right) \sinh^{2}{\left(3 x \right)}} - \frac{2 \log{\left(\sinh{\left(3 x \right)} \right)}}{\left(\left(x + 2\right)^{2} + 1\right)^{2}}\right) \sinh^{\operatorname{atan}{\left(x + 2 \right)}}{\left(3 x \right)}$$

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Derivative of sh3x^arctg(x+2)

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