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sin(2*x)^(2)

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

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

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Piecewise:

The solution

You have entered [src]
   2     
sin (2*x)
sin2(2x)\sin^{2}{\left(2 x \right)}
d /   2     \
--\sin (2*x)/
dx           
ddxsin2(2x)\frac{d}{d x} \sin^{2}{\left(2 x \right)}
Detail solution
  1. Let u=sin(2x)u = \sin{\left(2 x \right)}.

  2. Apply the power rule: u2u^{2} goes to 2u2 u

  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:

    4sin(2x)cos(2x)4 \sin{\left(2 x \right)} \cos{\left(2 x \right)}

  4. Now simplify:

    2sin(4x)2 \sin{\left(4 x \right)}


The answer is:

2sin(4x)2 \sin{\left(4 x \right)}

The graph
02468-8-6-4-2-10105-5
The first derivative [src]
4*cos(2*x)*sin(2*x)
4sin(2x)cos(2x)4 \sin{\left(2 x \right)} \cos{\left(2 x \right)}
The second derivative [src]
  /   2           2     \
8*\cos (2*x) - sin (2*x)/
8(sin2(2x)+cos2(2x))8 \left(- \sin^{2}{\left(2 x \right)} + \cos^{2}{\left(2 x \right)}\right)
The third derivative [src]
-64*cos(2*x)*sin(2*x)
64sin(2x)cos(2x)- 64 \sin{\left(2 x \right)} \cos{\left(2 x \right)}
The graph
Derivative of sin(2*x)^(2)