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(tan(x))^sin(x)
The derivative of the function/ (tan(x))^sin(x)

Derivative of (tan(x))^sin(x)

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

The graph:

from to

Piecewise:

The solution

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

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

    But the derivative is

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

The answer is:

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

The graph
02468-8-6-4-2-1010-10001000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
             /                     /       2   \       \
   sin(x)    |                     \1 + tan (x)/*sin(x)|
tan      (x)*|cos(x)*log(tan(x)) + --------------------|
             \                            tan(x)       /
((tan⁡2(x)+1)sin⁡(x)tan⁡(x)+log⁡(tan⁡(x))cos⁡(x))tan⁡sin⁡(x)(x)\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)} \cos{\left(x \right)}\right) \tan^{\sin{\left(x \right)}}{\left(x \right)}
Simplify
The second derivative [src] 
             /                                           2                                                              2                                \
             |/                     /       2   \       \                                                  /       2   \             /       2   \       |
   sin(x)    ||                     \1 + tan (x)/*sin(x)|                           /       2   \          \1 + tan (x)/ *sin(x)   2*\1 + tan (x)/*cos(x)|
tan      (x)*||cos(x)*log(tan(x)) + --------------------|  - log(tan(x))*sin(x) + 2*\1 + tan (x)/*sin(x) - --------------------- + ----------------------|
             |\                            tan(x)       /                                                            2                     tan(x)        |
             \                                                                                                    tan (x)                                /
(((tan⁡2(x)+1)sin⁡(x)tan⁡(x)+log⁡(tan⁡(x))cos⁡(x))2−(tan⁡2(x)+1)2sin⁡(x)tan⁡2(x)+2(tan⁡2(x)+1)sin⁡(x)+2(tan⁡2(x)+1)cos⁡(x)tan⁡(x)−log⁡(tan⁡(x))sin⁡(x))tan⁡sin⁡(x)(x)\left(\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)} \cos{\left(x \right)}\right)^{2} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\tan^{2}{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)}} - \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}\right) \tan^{\sin{\left(x \right)}}{\left(x \right)}
Simplify
The third derivative [src] 
             /                                           3                                                                      /                                                           2                                \                                           2                         2                                                  3                                       \
             |/                     /       2   \       \                           /                     /       2   \       \ |                                              /       2   \             /       2   \       |                              /       2   \             /       2   \             /       2   \            /       2   \                                        |
   sin(x)    ||                     \1 + tan (x)/*sin(x)|                           |                     \1 + tan (x)/*sin(x)| |                       /       2   \          \1 + tan (x)/ *sin(x)   2*\1 + tan (x)/*cos(x)|     /       2   \          4*\1 + tan (x)/ *sin(x)   3*\1 + tan (x)/ *cos(x)   3*\1 + tan (x)/*sin(x)   2*\1 + tan (x)/ *sin(x)     /       2   \              |
tan      (x)*||cos(x)*log(tan(x)) + --------------------|  - cos(x)*log(tan(x)) - 3*|cos(x)*log(tan(x)) + --------------------|*|log(tan(x))*sin(x) - 2*\1 + tan (x)/*sin(x) + --------------------- - ----------------------| + 6*\1 + tan (x)/*cos(x) - ----------------------- - ----------------------- - ---------------------- + ----------------------- + 4*\1 + tan (x)/*sin(x)*tan(x)|
             |\                            tan(x)       /                           \                            tan(x)       / |                                                        2                     tan(x)        |                                     tan(x)                      2                      tan(x)                      3                                           |
             \                                                                                                                  \                                                     tan (x)                                /                                                              tan (x)                                            tan (x)                                        /
(((tan⁡2(x)+1)sin⁡(x)tan⁡(x)+log⁡(tan⁡(x))cos⁡(x))3−3((tan⁡2(x)+1)sin⁡(x)tan⁡(x)+log⁡(tan⁡(x))cos⁡(x))((tan⁡2(x)+1)2sin⁡(x)tan⁡2(x)−2(tan⁡2(x)+1)sin⁡(x)−2(tan⁡2(x)+1)cos⁡(x)tan⁡(x)+log⁡(tan⁡(x))sin⁡(x))+2(tan⁡2(x)+1)3sin⁡(x)tan⁡3(x)−4(tan⁡2(x)+1)2sin⁡(x)tan⁡(x)−3(tan⁡2(x)+1)2cos⁡(x)tan⁡2(x)+4(tan⁡2(x)+1)sin⁡(x)tan⁡(x)−3(tan⁡2(x)+1)sin⁡(x)tan⁡(x)+6(tan⁡2(x)+1)cos⁡(x)−log⁡(tan⁡(x))cos⁡(x))tan⁡sin⁡(x)(x)\left(\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)} \cos{\left(x \right)}\right)^{3} - 3 \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)} \cos{\left(x \right)}\right) \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\tan^{2}{\left(x \right)}} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}\right) + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \sin{\left(x \right)}}{\tan^{3}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\tan{\left(x \right)}} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)}}{\tan^{2}{\left(x \right)}} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)}} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \log{\left(\tan{\left(x \right)} \right)} \cos{\left(x \right)}\right) \tan^{\sin{\left(x \right)}}{\left(x \right)}
Simplify
The graph
Derivative of (tan(x))^sin(x)
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