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y(x)=3sin^2(2x)

Derivative of y(x)=3sin^2(2x)

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

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The solution

You have entered [src]
     2     
3*sin (2*x)
3sin2(2x)3 \sin^{2}{\left(2 x \right)}
Detail solution
  1. The derivative of a constant times a function is the constant times the derivative of the function.

    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)}

    So, the result is: 12sin(2x)cos(2x)12 \sin{\left(2 x \right)} \cos{\left(2 x \right)}

  2. Now simplify:

    6sin(4x)6 \sin{\left(4 x \right)}


The answer is:

6sin(4x)6 \sin{\left(4 x \right)}

The graph
02468-8-6-4-2-1010-1010
The first derivative [src]
12*cos(2*x)*sin(2*x)
12sin(2x)cos(2x)12 \sin{\left(2 x \right)} \cos{\left(2 x \right)}
The second derivative [src]
   /   2           2     \
24*\cos (2*x) - sin (2*x)/
24(sin2(2x)+cos2(2x))24 \left(- \sin^{2}{\left(2 x \right)} + \cos^{2}{\left(2 x \right)}\right)
The third derivative [src]
-192*cos(2*x)*sin(2*x)
192sin(2x)cos(2x)- 192 \sin{\left(2 x \right)} \cos{\left(2 x \right)}
The graph
Derivative of y(x)=3sin^2(2x)