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The derivative of the function/ (sinx)^(arccos(2*x))

Derivative of (sinx)^(arccos(2*x))

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

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Piecewise:

The solution

You have entered [src] 
   acos(2*x)   
sin         (x)
sin⁡acos⁡(2x)(x)\sin^{\operatorname{acos}{\left(2 x \right)}}{\left(x \right)} 
sin(x)^acos(2*x)
Detail solution

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

    But the derivative is

    (log⁡(acos⁡(2x))+1)acos⁡acos⁡(2x)(2x)\left(\log{\left(\operatorname{acos}{\left(2 x \right)} \right)} + 1\right) \operatorname{acos}^{\operatorname{acos}{\left(2 x \right)}}{\left(2 x \right)}

The answer is:

(log⁡(acos⁡(2x))+1)acos⁡acos⁡(2x)(2x)\left(\log{\left(\operatorname{acos}{\left(2 x \right)} \right)} + 1\right) \operatorname{acos}^{\operatorname{acos}{\left(2 x \right)}}{\left(2 x \right)}

The graph
02468-8-6-4-2-101004
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   acos(2*x)    /  2*log(sin(x))   acos(2*x)*cos(x)\
sin         (x)*|- ------------- + ----------------|
                |     __________        sin(x)     |
                |    /        2                    |
                \  \/  1 - 4*x                     /
(cos⁡(x)acos⁡(2x)sin⁡(x)−2log⁡(sin⁡(x))1−4x2)sin⁡acos⁡(2x)(x)\left(\frac{\cos{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{1 - 4 x^{2}}}\right) \sin^{\operatorname{acos}{\left(2 x \right)}}{\left(x \right)}
Simplify
The second derivative [src] 
                /                                    2                  2                                                      \
   acos(2*x)    |/  2*log(sin(x))   acos(2*x)*cos(x)\                cos (x)*acos(2*x)   8*x*log(sin(x))         4*cos(x)      |
sin         (x)*||- ------------- + ----------------|  - acos(2*x) - ----------------- - --------------- - --------------------|
                ||     __________        sin(x)     |                        2                      3/2       __________       |
                ||    /        2                    |                     sin (x)         /       2\         /        2        |
                \\  \/  1 - 4*x                     /                                     \1 - 4*x /       \/  1 - 4*x  *sin(x)/
(−8xlog⁡(sin⁡(x))(1−4x2)32+(cos⁡(x)acos⁡(2x)sin⁡(x)−2log⁡(sin⁡(x))1−4x2)2−acos⁡(2x)−cos⁡2(x)acos⁡(2x)sin⁡2(x)−4cos⁡(x)1−4x2sin⁡(x))sin⁡acos⁡(2x)(x)\left(- \frac{8 x \log{\left(\sin{\left(x \right)} \right)}}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}} + \left(\frac{\cos{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{1 - 4 x^{2}}}\right)^{2} - \operatorname{acos}{\left(2 x \right)} - \frac{\cos^{2}{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin^{2}{\left(x \right)}} - \frac{4 \cos{\left(x \right)}}{\sqrt{1 - 4 x^{2}} \sin{\left(x \right)}}\right) \sin^{\operatorname{acos}{\left(2 x \right)}}{\left(x \right)}
Simplify
The third derivative [src] 
                /                                    3                                                                          /   2                                                                  \       2                    3                                                2                                \
   acos(2*x)    |/  2*log(sin(x))   acos(2*x)*cos(x)\          6         8*log(sin(x))     /  2*log(sin(x))   acos(2*x)*cos(x)\ |cos (x)*acos(2*x)         4*cos(x)         8*x*log(sin(x))            |   96*x *log(sin(x))   2*cos (x)*acos(2*x)   2*acos(2*x)*cos(x)         6*cos (x)             24*x*cos(x)     |
sin         (x)*||- ------------- + ----------------|  + ------------- - ------------- - 3*|- ------------- + ----------------|*|----------------- + -------------------- + --------------- + acos(2*x)| - ----------------- + ------------------- + ------------------ + --------------------- - --------------------|
                ||     __________        sin(x)     |       __________             3/2     |     __________        sin(x)     | |        2              __________                     3/2             |               5/2              3                  sin(x)            __________                     3/2       |
                ||    /        2                    |      /        2    /       2\        |    /        2                    | |     sin (x)          /        2            /       2\                |     /       2\              sin (x)                                /        2     2      /       2\          |
                \\  \/  1 - 4*x                     /    \/  1 - 4*x     \1 - 4*x /        \  \/  1 - 4*x                     / \                    \/  1 - 4*x  *sin(x)    \1 - 4*x /                /     \1 - 4*x /                                                   \/  1 - 4*x  *sin (x)   \1 - 4*x /   *sin(x)/
(−96x2log⁡(sin⁡(x))(1−4x2)52−24xcos⁡(x)(1−4x2)32sin⁡(x)+(cos⁡(x)acos⁡(2x)sin⁡(x)−2log⁡(sin⁡(x))1−4x2)3−3(cos⁡(x)acos⁡(2x)sin⁡(x)−2log⁡(sin⁡(x))1−4x2)(8xlog⁡(sin⁡(x))(1−4x2)32+acos⁡(2x)+cos⁡2(x)acos⁡(2x)sin⁡2(x)+4cos⁡(x)1−4x2sin⁡(x))+2cos⁡(x)acos⁡(2x)sin⁡(x)+2cos⁡3(x)acos⁡(2x)sin⁡3(x)+61−4x2+6cos⁡2(x)1−4x2sin⁡2(x)−8log⁡(sin⁡(x))(1−4x2)32)sin⁡acos⁡(2x)(x)\left(- \frac{96 x^{2} \log{\left(\sin{\left(x \right)} \right)}}{\left(1 - 4 x^{2}\right)^{\frac{5}{2}}} - \frac{24 x \cos{\left(x \right)}}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}} \sin{\left(x \right)}} + \left(\frac{\cos{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{1 - 4 x^{2}}}\right)^{3} - 3 \left(\frac{\cos{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{1 - 4 x^{2}}}\right) \left(\frac{8 x \log{\left(\sin{\left(x \right)} \right)}}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}} + \operatorname{acos}{\left(2 x \right)} + \frac{\cos^{2}{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin^{2}{\left(x \right)}} + \frac{4 \cos{\left(x \right)}}{\sqrt{1 - 4 x^{2}} \sin{\left(x \right)}}\right) + \frac{2 \cos{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin{\left(x \right)}} + \frac{2 \cos^{3}{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin^{3}{\left(x \right)}} + \frac{6}{\sqrt{1 - 4 x^{2}}} + \frac{6 \cos^{2}{\left(x \right)}}{\sqrt{1 - 4 x^{2}} \sin^{2}{\left(x \right)}} - \frac{8 \log{\left(\sin{\left(x \right)} \right)}}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}}\right) \sin^{\operatorname{acos}{\left(2 x \right)}}{\left(x \right)}
Simplify
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