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sin(2*x)^(3)

Derivative of sin(2*x)^(3)

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

You have entered [src]
   3     
sin (2*x)
sin3(2x)\sin^{3}{\left(2 x \right)}
sin(2*x)^3
Detail solution
  1. Let u=sin(2x)u = \sin{\left(2 x \right)}.

  2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

  3. Then, apply the chain rule. Multiply by ddxsin(2x)\frac{d}{d x} \sin{\left(2 x \right)}:

    1. Let u=2xu = 2 x.

    2. The derivative of sine is cosine:

      ddusin(u)=cos(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx2x\frac{d}{d x} 2 x:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: xx goes to 11

        So, the result is: 22

      The result of the chain rule is:

      2cos(2x)2 \cos{\left(2 x \right)}

    The result of the chain rule is:

    6sin2(2x)cos(2x)6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)}


The answer is:

6sin2(2x)cos(2x)6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)}

The graph
02468-8-6-4-2-10105-5
The first derivative [src]
     2              
6*sin (2*x)*cos(2*x)
6sin2(2x)cos(2x)6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)}
The second derivative [src]
   /     2             2     \         
12*\- sin (2*x) + 2*cos (2*x)/*sin(2*x)
12(sin2(2x)+2cos2(2x))sin(2x)12 \left(- \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)}
The third derivative [src]
   /       2             2     \         
24*\- 7*sin (2*x) + 2*cos (2*x)/*cos(2*x)
24(7sin2(2x)+2cos2(2x))cos(2x)24 \left(- 7 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \cos{\left(2 x \right)}
The graph
Derivative of sin(2*x)^(3)