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(sin(3*x))^acos(x)
The derivative of the function/ (sin(3*x))^acos(x)

Derivative of (sin(3*x))^acos(x)

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

from to

Piecewise:

The solution

You have entered [src] 
   acos(x)     
sin       (3*x)
sin⁡acos⁡(x)(3x)\sin^{\operatorname{acos}{\left(x \right)}}{\left(3 x \right)} 
sin(3*x)^acos(x)
Detail solution

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

    But the derivative is

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

The answer is:

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

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   acos(x)      /  log(sin(3*x))   3*acos(x)*cos(3*x)\
sin       (3*x)*|- ------------- + ------------------|
                |      ________         sin(3*x)     |
                |     /      2                       |
                \   \/  1 - x                        /
(3cos⁡(3x)acos⁡(x)sin⁡(3x)−log⁡(sin⁡(3x))1−x2)sin⁡acos⁡(x)(3x)\left(\frac{3 \cos{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{1 - x^{2}}}\right) \sin^{\operatorname{acos}{\left(x \right)}}{\left(3 x \right)}
Simplify
The second derivative [src] 
                /                                      2                                      2                                    \
   acos(x)      |/  log(sin(3*x))   3*acos(x)*cos(3*x)\                x*log(sin(3*x))   9*cos (3*x)*acos(x)        6*cos(3*x)     |
sin       (3*x)*||- ------------- + ------------------|  - 9*acos(x) - --------------- - ------------------- - --------------------|
                ||      ________         sin(3*x)     |                          3/2             2                ________         |
                ||     /      2                       |                  /     2\             sin (3*x)          /      2          |
                \\   \/  1 - x                        /                  \1 - x /                              \/  1 - x  *sin(3*x)/
(−xlog⁡(sin⁡(3x))(1−x2)32+(3cos⁡(3x)acos⁡(x)sin⁡(3x)−log⁡(sin⁡(3x))1−x2)2−9acos⁡(x)−9cos⁡2(3x)acos⁡(x)sin⁡2(3x)−6cos⁡(3x)1−x2sin⁡(3x))sin⁡acos⁡(x)(3x)\left(- \frac{x \log{\left(\sin{\left(3 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \left(\frac{3 \cos{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{1 - x^{2}}}\right)^{2} - 9 \operatorname{acos}{\left(x \right)} - \frac{9 \cos^{2}{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin^{2}{\left(3 x \right)}} - \frac{6 \cos{\left(3 x \right)}}{\sqrt{1 - x^{2}} \sin{\left(3 x \right)}}\right) \sin^{\operatorname{acos}{\left(x \right)}}{\left(3 x \right)}
Simplify
The third derivative [src] 
                /                                      3                                                                          /                                                          2             \      2                            2                  3                                                          \
   acos(x)      |/  log(sin(3*x))   3*acos(x)*cos(3*x)\         27       log(sin(3*x))     /  log(sin(3*x))   3*acos(x)*cos(3*x)\ |            x*log(sin(3*x))        6*cos(3*x)        9*cos (3*x)*acos(x)|   3*x *log(sin(3*x))        27*cos (3*x)       54*cos (3*x)*acos(x)   54*acos(x)*cos(3*x)       9*x*cos(3*x)    |
sin       (3*x)*||- ------------- + ------------------|  + ----------- - ------------- - 3*|- ------------- + ------------------|*|9*acos(x) + --------------- + -------------------- + -------------------| - ------------------ + --------------------- + -------------------- + ------------------- - --------------------|
                ||      ________         sin(3*x)     |       ________            3/2      |      ________         sin(3*x)     | |                      3/2        ________                    2          |              5/2          ________                     3                    sin(3*x)                3/2         |
                ||     /      2                       |      /      2     /     2\         |     /      2                       | |              /     2\          /      2                  sin (3*x)     |      /     2\            /      2     2             sin (3*x)                               /     2\            |
                \\   \/  1 - x                        /    \/  1 - x      \1 - x /         \   \/  1 - x                        / \              \1 - x /        \/  1 - x  *sin(3*x)                      /      \1 - x /          \/  1 - x  *sin (3*x)                                                \1 - x /   *sin(3*x)/
(−3x2log⁡(sin⁡(3x))(1−x2)52−9xcos⁡(3x)(1−x2)32sin⁡(3x)+(3cos⁡(3x)acos⁡(x)sin⁡(3x)−log⁡(sin⁡(3x))1−x2)3−3(3cos⁡(3x)acos⁡(x)sin⁡(3x)−log⁡(sin⁡(3x))1−x2)(xlog⁡(sin⁡(3x))(1−x2)32+9acos⁡(x)+9cos⁡2(3x)acos⁡(x)sin⁡2(3x)+6cos⁡(3x)1−x2sin⁡(3x))+54cos⁡(3x)acos⁡(x)sin⁡(3x)+54cos⁡3(3x)acos⁡(x)sin⁡3(3x)+271−x2+27cos⁡2(3x)1−x2sin⁡2(3x)−log⁡(sin⁡(3x))(1−x2)32)sin⁡acos⁡(x)(3x)\left(- \frac{3 x^{2} \log{\left(\sin{\left(3 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} - \frac{9 x \cos{\left(3 x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}} \sin{\left(3 x \right)}} + \left(\frac{3 \cos{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{1 - x^{2}}}\right)^{3} - 3 \left(\frac{3 \cos{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{1 - x^{2}}}\right) \left(\frac{x \log{\left(\sin{\left(3 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + 9 \operatorname{acos}{\left(x \right)} + \frac{9 \cos^{2}{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin^{2}{\left(3 x \right)}} + \frac{6 \cos{\left(3 x \right)}}{\sqrt{1 - x^{2}} \sin{\left(3 x \right)}}\right) + \frac{54 \cos{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin{\left(3 x \right)}} + \frac{54 \cos^{3}{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin^{3}{\left(3 x \right)}} + \frac{27}{\sqrt{1 - x^{2}}} + \frac{27 \cos^{2}{\left(3 x \right)}}{\sqrt{1 - x^{2}} \sin^{2}{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}}\right) \sin^{\operatorname{acos}{\left(x \right)}}{\left(3 x \right)}
Simplify
The graph
Derivative of (sin(3*x))^acos(x)
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