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(x^4+5)^cot(x)
The derivative of the function/ (x^4+5)^cot(x)

Derivative of (x^4+5)^cot(x)

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

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The solution

You have entered [src] 
        cot(x)
/ 4    \      
\x  + 5/
(x4+5)cot⁡(x)\left(x^{4} + 5\right)^{\cot{\left(x \right)}} 
(x^4 + 5)^cot(x)
Detail solution

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

    But the derivative is

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

The answer is:

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

The graph
02468-8-6-4-2-1010-5e1995e199
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        cot(x) /                                3       \
/ 4    \       |/        2   \    / 4    \   4*x *cot(x)|
\x  + 5/      *|\-1 - cot (x)/*log\x  + 5/ + -----------|
               |                                 4      |
               \                                x  + 5  /
(x4+5)cot⁡(x)(4x3cot⁡(x)x4+5+(−cot⁡2(x)−1)log⁡(x4+5))\left(x^{4} + 5\right)^{\cot{\left(x \right)}} \left(\frac{4 x^{3} \cot{\left(x \right)}}{x^{4} + 5} + \left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(x^{4} + 5 \right)}\right)
Simplify
The second derivative [src] 
               /                                           2                                                                                        \
        cot(x) |/                                 3       \        6             3 /       2   \                                            2       |
/     4\       ||  /       2   \    /     4\   4*x *cot(x)|    16*x *cot(x)   8*x *\1 + cot (x)/     /       2   \           /     4\   12*x *cot(x)|
\5 + x /      *||- \1 + cot (x)/*log\5 + x / + -----------|  - ------------ - ------------------ + 2*\1 + cot (x)/*cot(x)*log\5 + x / + ------------|
               ||                                      4  |             2                4                                                      4   |
               |\                                 5 + x   /     /     4\            5 + x                                                  5 + x    |
               \                                                \5 + x /                                                                            /
(x4+5)cot⁡(x)(−16x6cot⁡(x)(x4+5)2−8x3(cot⁡2(x)+1)x4+5+12x2cot⁡(x)x4+5+(4x3cot⁡(x)x4+5−(cot⁡2(x)+1)log⁡(x4+5))2+2(cot⁡2(x)+1)log⁡(x4+5)cot⁡(x))\left(x^{4} + 5\right)^{\cot{\left(x \right)}} \left(- \frac{16 x^{6} \cot{\left(x \right)}}{\left(x^{4} + 5\right)^{2}} - \frac{8 x^{3} \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{4} + 5} + \frac{12 x^{2} \cot{\left(x \right)}}{x^{4} + 5} + \left(\frac{4 x^{3} \cot{\left(x \right)}}{x^{4} + 5} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{4} + 5 \right)}\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{4} + 5 \right)} \cot{\left(x \right)}\right)
Simplify
The third derivative [src] 
               /                                           3                                                                                                                                                                                                                                                                                                                                  \
        cot(x) |/                                 3       \      /                                 3       \ /                                        2             3 /       2   \      6       \                  2                    5              2 /       2   \                                                           6 /       2   \        9              3 /       2   \       |
/     4\       ||  /       2   \    /     4\   4*x *cot(x)|      |  /       2   \    /     4\   4*x *cot(x)| |  /       2   \           /     4\   6*x *cot(x)   4*x *\1 + cot (x)/   8*x *cot(x)|     /       2   \     /     4\   144*x *cot(x)   36*x *\1 + cot (x)/        2    /       2   \    /     4\   24*x*cot(x)   48*x *\1 + cot (x)/   128*x *cot(x)   24*x *\1 + cot (x)/*cot(x)|
\5 + x /      *||- \1 + cot (x)/*log\5 + x / + -----------|  - 6*|- \1 + cot (x)/*log\5 + x / + -----------|*|- \1 + cot (x)/*cot(x)*log\5 + x / - ----------- + ------------------ + -----------| - 2*\1 + cot (x)/ *log\5 + x / - ------------- - ------------------- - 4*cot (x)*\1 + cot (x)/*log\5 + x / + ----------- + ------------------- + ------------- + --------------------------|
               ||                                      4  |      |                                      4  | |                                             4                4                  2 |                                            2                 4                                                       4                  2                  3                    4          |
               |\                                 5 + x   /      \                                 5 + x   / |                                        5 + x            5 + x           /     4\  |                                    /     4\             5 + x                                                   5 + x           /     4\           /     4\                5 + x           |
               \                                                                                             \                                                                         \5 + x /  /                                    \5 + x /                                                                                     \5 + x /           \5 + x /                                /
(x4+5)cot⁡(x)(128x9cot⁡(x)(x4+5)3+48x6(cot⁡2(x)+1)(x4+5)2−144x5cot⁡(x)(x4+5)2+24x3(cot⁡2(x)+1)cot⁡(x)x4+5−36x2(cot⁡2(x)+1)x4+5+24xcot⁡(x)x4+5+(4x3cot⁡(x)x4+5−(cot⁡2(x)+1)log⁡(x4+5))3−6(4x3cot⁡(x)x4+5−(cot⁡2(x)+1)log⁡(x4+5))(8x6cot⁡(x)(x4+5)2+4x3(cot⁡2(x)+1)x4+5−6x2cot⁡(x)x4+5−(cot⁡2(x)+1)log⁡(x4+5)cot⁡(x))−2(cot⁡2(x)+1)2log⁡(x4+5)−4(cot⁡2(x)+1)log⁡(x4+5)cot⁡2(x))\left(x^{4} + 5\right)^{\cot{\left(x \right)}} \left(\frac{128 x^{9} \cot{\left(x \right)}}{\left(x^{4} + 5\right)^{3}} + \frac{48 x^{6} \left(\cot^{2}{\left(x \right)} + 1\right)}{\left(x^{4} + 5\right)^{2}} - \frac{144 x^{5} \cot{\left(x \right)}}{\left(x^{4} + 5\right)^{2}} + \frac{24 x^{3} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x^{4} + 5} - \frac{36 x^{2} \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{4} + 5} + \frac{24 x \cot{\left(x \right)}}{x^{4} + 5} + \left(\frac{4 x^{3} \cot{\left(x \right)}}{x^{4} + 5} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{4} + 5 \right)}\right)^{3} - 6 \left(\frac{4 x^{3} \cot{\left(x \right)}}{x^{4} + 5} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{4} + 5 \right)}\right) \left(\frac{8 x^{6} \cot{\left(x \right)}}{\left(x^{4} + 5\right)^{2}} + \frac{4 x^{3} \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{4} + 5} - \frac{6 x^{2} \cot{\left(x \right)}}{x^{4} + 5} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{4} + 5 \right)} \cot{\left(x \right)}\right) - 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x^{4} + 5 \right)} - 4 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{4} + 5 \right)} \cot^{2}{\left(x \right)}\right)
Simplify
The graph
Derivative of (x^4+5)^cot(x)
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