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y=(sqrtx)^tgx
The derivative of the function/ y=(sqrtx)^tgx

Derivative of y=(sqrtx)^tgx

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

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

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

    But the derivative is

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

The answer is:

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

The graph
02468-8-6-4-2-1010200000000-100000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 tan(x)                                    
 ------                                    
   2    //       2   \    /  ___\   tan(x)\
x      *|\1 + tan (x)/*log\\/ x / + ------|
        \                            2*x  /
xtan⁡(x)2((tan⁡2(x)+1)log⁡(x)+tan⁡(x)2x)x^{\frac{\tan{\left(x \right)}}{2}} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sqrt{x} \right)} + \frac{\tan{\left(x \right)}}{2 x}\right)
Simplify
The second derivative [src] 
 tan(x) /                       /tan(x)   /       2   \       \ /tan(x)     /       2   \    /  ___\\                                    \
 ------ |       2               |------ + \1 + tan (x)/*log(x)|*|------ + 2*\1 + tan (x)/*log\\/ x /|                                    |
   2    |1 + tan (x)   tan(x)   \  x                          / \  x                                /     /       2   \    /  ___\       |
x      *|----------- - ------ + --------------------------------------------------------------------- + 2*\1 + tan (x)/*log\\/ x /*tan(x)|
        |     x            2                                      4                                                                      |
        \               2*x                                                                                                              /
xtan⁡(x)2((2(tan⁡2(x)+1)log⁡(x)+tan⁡(x)x)((tan⁡2(x)+1)log⁡(x)+tan⁡(x)x)4+2(tan⁡2(x)+1)log⁡(x)tan⁡(x)+tan⁡2(x)+1x−tan⁡(x)2x2)x^{\frac{\tan{\left(x \right)}}{2}} \left(\frac{\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sqrt{x} \right)} + \frac{\tan{\left(x \right)}}{x}\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right)}{4} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sqrt{x} \right)} \tan{\left(x \right)} + \frac{\tan^{2}{\left(x \right)} + 1}{x} - \frac{\tan{\left(x \right)}}{2 x^{2}}\right)
Simplify
The third derivative [src] 
        /                                         /             /       2   \                                    \                                                                                         /             /       2   \                                \                                                                                                                                       \
        |         /tan(x)   /       2   \       \ |  tan(x)   2*\1 + tan (x)/     /       2   \    /  ___\       |                                                   /tan(x)     /       2   \    /  ___\\ |  tan(x)   2*\1 + tan (x)/     /       2   \              |                                  2                                                                                                    |
 tan(x) |         |------ + \1 + tan (x)/*log(x)|*|- ------ + --------------- + 4*\1 + tan (x)/*log\\/ x /*tan(x)|                                                   |------ + 2*\1 + tan (x)/*log\\/ x /|*|- ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)|   /tan(x)   /       2   \       \  /tan(x)     /       2   \    /  ___\\                                                              |
 ------ |         \  x                          / |     2            x                                           |                  2                /       2   \   \  x                                / |     2            x                                       |   |------ + \1 + tan (x)/*log(x)| *|------ + 2*\1 + tan (x)/*log\\/ x /|     /       2   \                                            |
   2    |tan(x)                                   \    x                                                         /     /       2   \     /  ___\   3*\1 + tan (x)/                                         \    x                                                     /   \  x                          /  \  x                                /   3*\1 + tan (x)/*tan(x)        2    /       2   \    /  ___\|
x      *|------ + ------------------------------------------------------------------------------------------------ + 2*\1 + tan (x)/ *log\\/ x / - --------------- + -------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------- + ---------------------- + 4*tan (x)*\1 + tan (x)/*log\\/ x /|
        |   3                                                    2                                                                                          2                                                        4                                                                                      8                                                x                                                |
        \  x                                                                                                                                             2*x                                                                                                                                                                                                                                                  /
xtan⁡(x)2((2(tan⁡2(x)+1)log⁡(x)+tan⁡(x)x)((tan⁡2(x)+1)log⁡(x)+tan⁡(x)x)28+(2(tan⁡2(x)+1)log⁡(x)+tan⁡(x)x)(2(tan⁡2(x)+1)log⁡(x)tan⁡(x)+2(tan⁡2(x)+1)x−tan⁡(x)x2)4+((tan⁡2(x)+1)log⁡(x)+tan⁡(x)x)(4(tan⁡2(x)+1)log⁡(x)tan⁡(x)+2(tan⁡2(x)+1)x−tan⁡(x)x2)2+2(tan⁡2(x)+1)2log⁡(x)+4(tan⁡2(x)+1)log⁡(x)tan⁡2(x)+3(tan⁡2(x)+1)tan⁡(x)x−3(tan⁡2(x)+1)2x2+tan⁡(x)x3)x^{\frac{\tan{\left(x \right)}}{2}} \left(\frac{\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sqrt{x} \right)} + \frac{\tan{\left(x \right)}}{x}\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right)^{2}}{8} + \frac{\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sqrt{x} \right)} + \frac{\tan{\left(x \right)}}{x}\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}\right)}{4} + \frac{\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right) \left(4 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sqrt{x} \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}\right)}{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\sqrt{x} \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sqrt{x} \right)} \tan^{2}{\left(x \right)} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{2 x^{2}} + \frac{\tan{\left(x \right)}}{x^{3}}\right)
Simplify
The graph
Derivative of y=(sqrtx)^tgx
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