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y=sin^8(6x)*arctg2x^2
The derivative of the function/ y=sin^8(6x)*arctg2x^2

Derivative of y=sin^8(6x)*arctg2x^2

Function f() - derivative -N order at the point
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   8          2     
sin (6*x)*atan (2*x)
sin⁡8(6x)atan⁡2(2x)\sin^{8}{\left(6 x \right)} \operatorname{atan}^{2}{\left(2 x \right)} 
d /   8          2     \
--\sin (6*x)*atan (2*x)/
dx
ddxsin⁡8(6x)atan⁡2(2x)\frac{d}{d x} \sin^{8}{\left(6 x \right)} \operatorname{atan}^{2}{\left(2 x \right)}
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The graph
02468-8-6-4-2-1010-5050
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The first derivative [src] 
     8                                                  
4*sin (6*x)*atan(2*x)          2         7              
--------------------- + 48*atan (2*x)*sin (6*x)*cos(6*x)
              2                                         
       1 + 4*x
48sin⁡7(6x)cos⁡(6x)atan⁡2(2x)+4sin⁡8(6x)atan⁡(2x)4x2+148 \sin^{7}{\left(6 x \right)} \cos{\left(6 x \right)} \operatorname{atan}^{2}{\left(2 x \right)} + \frac{4 \sin^{8}{\left(6 x \right)} \operatorname{atan}{\left(2 x \right)}}{4 x^{2} + 1}
Simplify
The second derivative [src] 
            /                                               2                                                           \
     6      |         2      /   2             2     \   sin (6*x)*(-1 + 4*x*atan(2*x))   48*atan(2*x)*cos(6*x)*sin(6*x)|
8*sin (6*x)*|- 36*atan (2*x)*\sin (6*x) - 7*cos (6*x)/ - ------------------------------ + ------------------------------|
            |                                                               2                               2           |
            |                                                     /       2\                         1 + 4*x            |
            \                                                     \1 + 4*x /                                            /
8(−36(sin⁡2(6x)−7cos⁡2(6x))atan⁡2(2x)+48sin⁡(6x)cos⁡(6x)atan⁡(2x)4x2+1−(4xatan⁡(2x)−1)sin⁡2(6x)(4x2+1)2)sin⁡6(6x)8 \left(- 36 \left(\sin^{2}{\left(6 x \right)} - 7 \cos^{2}{\left(6 x \right)}\right) \operatorname{atan}^{2}{\left(2 x \right)} + \frac{48 \sin{\left(6 x \right)} \cos{\left(6 x \right)} \operatorname{atan}{\left(2 x \right)}}{4 x^{2} + 1} - \frac{\left(4 x \operatorname{atan}{\left(2 x \right)} - 1\right) \sin^{2}{\left(6 x \right)}}{\left(4 x^{2} + 1\right)^{2}}\right) \sin^{6}{\left(6 x \right)}
Simplify
The third derivative [src] 
              /          /               2                      \                                                                                                                                                          \
              |   3      |  6*x      16*x *atan(2*x)            |                                                                                                                                                          |
              |sin (6*x)*|-------- - --------------- + atan(2*x)|                                                                                                                                                          |
              |          |       2              2               |                                                                   2                                          /   2             2     \                   |
       5      |          \1 + 4*x        1 + 4*x                /           2      /        2              2     \            36*sin (6*x)*(-1 + 4*x*atan(2*x))*cos(6*x)   108*\sin (6*x) - 7*cos (6*x)/*atan(2*x)*sin(6*x)|
-32*sin (6*x)*|-------------------------------------------------- + 108*atan (2*x)*\- 21*cos (6*x) + 11*sin (6*x)/*cos(6*x) + ------------------------------------------ + ------------------------------------------------|
              |                             2                                                                                                          2                                              2                    |
              |                   /       2\                                                                                                 /       2\                                        1 + 4*x                     |
              \                   \1 + 4*x /                                                                                                 \1 + 4*x /                                                                    /
−32⋅(108⋅(11sin⁡2(6x)−21cos⁡2(6x))cos⁡(6x)atan⁡2(2x)+108(sin⁡2(6x)−7cos⁡2(6x))sin⁡(6x)atan⁡(2x)4x2+1+36⋅(4xatan⁡(2x)−1)sin⁡2(6x)cos⁡(6x)(4x2+1)2+(−16x2atan⁡(2x)4x2+1+6x4x2+1+atan⁡(2x))sin⁡3(6x)(4x2+1)2)sin⁡5(6x)- 32 \cdot \left(108 \cdot \left(11 \sin^{2}{\left(6 x \right)} - 21 \cos^{2}{\left(6 x \right)}\right) \cos{\left(6 x \right)} \operatorname{atan}^{2}{\left(2 x \right)} + \frac{108 \left(\sin^{2}{\left(6 x \right)} - 7 \cos^{2}{\left(6 x \right)}\right) \sin{\left(6 x \right)} \operatorname{atan}{\left(2 x \right)}}{4 x^{2} + 1} + \frac{36 \cdot \left(4 x \operatorname{atan}{\left(2 x \right)} - 1\right) \sin^{2}{\left(6 x \right)} \cos{\left(6 x \right)}}{\left(4 x^{2} + 1\right)^{2}} + \frac{\left(- \frac{16 x^{2} \operatorname{atan}{\left(2 x \right)}}{4 x^{2} + 1} + \frac{6 x}{4 x^{2} + 1} + \operatorname{atan}{\left(2 x \right)}\right) \sin^{3}{\left(6 x \right)}}{\left(4 x^{2} + 1\right)^{2}}\right) \sin^{5}{\left(6 x \right)}
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Derivative of y=sin^8(6x)*arctg2x^2
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