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The derivative of the function/ (tg(sqrt(x)))^(ln(x))

Derivative of (tg(sqrt(x)))^(ln(x))

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   log(x)/  ___\
tan      \\/ x /
tan⁡log⁡(x)(x)\tan^{\log{\left(x \right)}}{\left(\sqrt{x} \right)} 
tan(sqrt(x))^log(x)
Detail solution

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

    But the derivative is

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

The answer is:

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

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                 /   /   /  ___\\   /       2/  ___\\       \
   log(x)/  ___\ |log\tan\\/ x //   \1 + tan \\/ x //*log(x)|
tan      \\/ x /*|--------------- + ------------------------|
                 |       x                 ___    /  ___\   |
                 \                     2*\/ x *tan\\/ x /   /
(log⁡(tan⁡(x))x+(tan⁡2(x)+1)log⁡(x)2xtan⁡(x))tan⁡log⁡(x)(x)\left(\frac{\log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{2 \sqrt{x} \tan{\left(\sqrt{x} \right)}}\right) \tan^{\log{\left(x \right)}}{\left(\sqrt{x} \right)}
Simplify
The second derivative [src] 
                 /                                              2                                                                                                                      \
                 |/     /   /  ___\\   /       2/  ___\\       \                                                                                                                       |
                 ||2*log\tan\\/ x //   \1 + tan \\/ x //*log(x)|                                                                                                                       |
                 ||----------------- + ------------------------|                                                                                    2                                  |
                 ||        x                 ___    /  ___\    |       /   /  ___\\          2/  ___\   /       2/  ___\\          /       2/  ___\\           /       2/  ___\\       |
   log(x)/  ___\ |\                        \/ x *tan\\/ x /    /    log\tan\\/ x //   1 + tan \\/ x /   \1 + tan \\/ x //*log(x)   \1 + tan \\/ x // *log(x)   \1 + tan \\/ x //*log(x)|
tan      \\/ x /*|----------------------------------------------- - --------------- + --------------- + ------------------------ - ------------------------- - ------------------------|
                 |                       4                                  2          3/2    /  ___\             2*x                          2/  ___\              3/2    /  ___\    |
                 \                                                         x          x   *tan\\/ x /                                   4*x*tan \\/ x /           4*x   *tan\\/ x /    /
((2log⁡(tan⁡(x))x+(tan⁡2(x)+1)log⁡(x)xtan⁡(x))24−(tan⁡2(x)+1)2log⁡(x)4xtan⁡2(x)+(tan⁡2(x)+1)log⁡(x)2x−log⁡(tan⁡(x))x2−(tan⁡2(x)+1)log⁡(x)4x32tan⁡(x)+tan⁡2(x)+1x32tan⁡(x))tan⁡log⁡(x)(x)\left(\frac{\left(\frac{2 \log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{\sqrt{x} \tan{\left(\sqrt{x} \right)}}\right)^{2}}{4} - \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{2} \log{\left(x \right)}}{4 x \tan^{2}{\left(\sqrt{x} \right)}} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{2 x} - \frac{\log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x^{2}} - \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{4 x^{\frac{3}{2}} \tan{\left(\sqrt{x} \right)}} + \frac{\tan^{2}{\left(\sqrt{x} \right)} + 1}{x^{\frac{3}{2}} \tan{\left(\sqrt{x} \right)}}\right) \tan^{\log{\left(x \right)}}{\left(\sqrt{x} \right)}
Simplify
The third derivative [src] 
                 /                                              3                                                                        /                                                                                        2                                  \                                                                                                                                                                                                                                                         \
                 |/     /   /  ___\\   /       2/  ___\\       \                          /     /   /  ___\\   /       2/  ___\\       \ |     /   /  ___\\     /       2/  ___\\     /       2/  ___\\          /       2/  ___\\           /       2/  ___\\       |                                                                                                                                                                                                                                                         |
                 ||2*log\tan\\/ x //   \1 + tan \\/ x //*log(x)|                          |2*log\tan\\/ x //   \1 + tan \\/ x //*log(x)| |4*log\tan\\/ x //   4*\1 + tan \\/ x //   2*\1 + tan \\/ x //*log(x)   \1 + tan \\/ x // *log(x)   \1 + tan \\/ x //*log(x)|                                                                                                                                                                                                                                                         |
                 ||----------------- + ------------------------|                        3*|----------------- + ------------------------|*|----------------- - ------------------- - -------------------------- + ------------------------- + ------------------------|                                                                  2                                                                                       2                           3                             2                                    |
                 ||        x                 ___    /  ___\    |         /   /  ___\\     |        x                 ___    /  ___\    | |         2             3/2    /  ___\                 x                           2/  ___\              3/2    /  ___\     |     /       2/  ___\\     /       2/  ___\\     /       2/  ___\\      /       2/  ___\\          /       2/  ___\\           /  ___\   /       2/  ___\\           /       2/  ___\\             /       2/  ___\\             /       2/  ___\\       |
   log(x)/  ___\ |\                        \/ x *tan\\/ x /    /    2*log\tan\\/ x //     \                        \/ x *tan\\/ x /    / \        x             x   *tan\\/ x /                                        x*tan \\/ x /             x   *tan\\/ x /     /   3*\1 + tan \\/ x //   9*\1 + tan \\/ x //   3*\1 + tan \\/ x //    3*\1 + tan \\/ x //*log(x)   \1 + tan \\/ x //*log(x)*tan\\/ x /   \1 + tan \\/ x // *log(x)   \1 + tan \\/ x // *log(x)   3*\1 + tan \\/ x // *log(x)   3*\1 + tan \\/ x //*log(x)|
tan      \\/ x /*|----------------------------------------------- + ----------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + ------------------- - ------------------- - -------------------- - -------------------------- + ----------------------------------- - ------------------------- + ------------------------- + --------------------------- + --------------------------|
                 |                       8                                   3                                                                                                8                                                                                                     2              5/2    /  ___\         2    2/  ___\                   2                                3/2                        3/2    /  ___\              3/2    3/  ___\               2    2/  ___\               5/2    /  ___\     |
                 \                                                          x                                                                                                                                                                                                    2*x            4*x   *tan\\/ x /      4*x *tan \\/ x /                4*x                              2*x                        2*x   *tan\\/ x /           4*x   *tan \\/ x /            8*x *tan \\/ x /            8*x   *tan\\/ x /     /
((2log⁡(tan⁡(x))x+(tan⁡2(x)+1)log⁡(x)xtan⁡(x))38−3(2log⁡(tan⁡(x))x+(tan⁡2(x)+1)log⁡(x)xtan⁡(x))((tan⁡2(x)+1)2log⁡(x)xtan⁡2(x)−2(tan⁡2(x)+1)log⁡(x)x+4log⁡(tan⁡(x))x2+(tan⁡2(x)+1)log⁡(x)x32tan⁡(x)−4(tan⁡2(x)+1)x32tan⁡(x))8+3(tan⁡2(x)+1)2log⁡(x)8x2tan⁡2(x)−3(tan⁡2(x)+1)24x2tan⁡2(x)−3(tan⁡2(x)+1)log⁡(x)4x2+3(tan⁡2(x)+1)2x2+2log⁡(tan⁡(x))x3+(tan⁡2(x)+1)3log⁡(x)4x32tan⁡3(x)−(tan⁡2(x)+1)2log⁡(x)2x32tan⁡(x)+(tan⁡2(x)+1)log⁡(x)tan⁡(x)2x32+3(tan⁡2(x)+1)log⁡(x)8x52tan⁡(x)−9(tan⁡2(x)+1)4x52tan⁡(x))tan⁡log⁡(x)(x)\left(\frac{\left(\frac{2 \log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{\sqrt{x} \tan{\left(\sqrt{x} \right)}}\right)^{3}}{8} - \frac{3 \left(\frac{2 \log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{\sqrt{x} \tan{\left(\sqrt{x} \right)}}\right) \left(\frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{2} \log{\left(x \right)}}{x \tan^{2}{\left(\sqrt{x} \right)}} - \frac{2 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{x} + \frac{4 \log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x^{2}} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{x^{\frac{3}{2}} \tan{\left(\sqrt{x} \right)}} - \frac{4 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)}{x^{\frac{3}{2}} \tan{\left(\sqrt{x} \right)}}\right)}{8} + \frac{3 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{2} \log{\left(x \right)}}{8 x^{2} \tan^{2}{\left(\sqrt{x} \right)}} - \frac{3 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{2}}{4 x^{2} \tan^{2}{\left(\sqrt{x} \right)}} - \frac{3 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{4 x^{2}} + \frac{3 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)}{2 x^{2}} + \frac{2 \log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x^{3}} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{3} \log{\left(x \right)}}{4 x^{\frac{3}{2}} \tan^{3}{\left(\sqrt{x} \right)}} - \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{2} \log{\left(x \right)}}{2 x^{\frac{3}{2}} \tan{\left(\sqrt{x} \right)}} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)} \tan{\left(\sqrt{x} \right)}}{2 x^{\frac{3}{2}}} + \frac{3 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{8 x^{\frac{5}{2}} \tan{\left(\sqrt{x} \right)}} - \frac{9 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)}{4 x^{\frac{5}{2}} \tan{\left(\sqrt{x} \right)}}\right) \tan^{\log{\left(x \right)}}{\left(\sqrt{x} \right)}
Simplify
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