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sin^3(2x+1)

Derivative of sin^3(2x+1)

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

The solution

You have entered [src]
   3         
sin (2*x + 1)
sin3(2x+1)\sin^{3}{\left(2 x + 1 \right)}
d /   3         \
--\sin (2*x + 1)/
dx               
ddxsin3(2x+1)\frac{d}{d x} \sin^{3}{\left(2 x + 1 \right)}
Detail solution
  1. Let u=sin(2x+1)u = \sin{\left(2 x + 1 \right)}.

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

  3. Then, apply the chain rule. Multiply by ddxsin(2x+1)\frac{d}{d x} \sin{\left(2 x + 1 \right)}:

    1. Let u=2x+1u = 2 x + 1.

    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 ddx(2x+1)\frac{d}{d x} \left(2 x + 1\right):

      1. Differentiate 2x+12 x + 1 term by term:

        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

        2. The derivative of the constant 11 is zero.

        The result is: 22

      The result of the chain rule is:

      2cos(2x+1)2 \cos{\left(2 x + 1 \right)}

    The result of the chain rule is:

    6sin2(2x+1)cos(2x+1)6 \sin^{2}{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}

  4. Now simplify:

    6sin2(2x+1)cos(2x+1)6 \sin^{2}{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}


The answer is:

6sin2(2x+1)cos(2x+1)6 \sin^{2}{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}

The graph
02468-8-6-4-2-10105-5
The first derivative [src]
     2                      
6*sin (2*x + 1)*cos(2*x + 1)
6sin2(2x+1)cos(2x+1)6 \sin^{2}{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}
The second derivative [src]
   /     2                 2         \             
12*\- sin (1 + 2*x) + 2*cos (1 + 2*x)/*sin(1 + 2*x)
12(sin2(2x+1)+2cos2(2x+1))sin(2x+1)12 \left(- \sin^{2}{\left(2 x + 1 \right)} + 2 \cos^{2}{\left(2 x + 1 \right)}\right) \sin{\left(2 x + 1 \right)}
The third derivative [src]
   /       2                 2         \             
24*\- 7*sin (1 + 2*x) + 2*cos (1 + 2*x)/*cos(1 + 2*x)
24(7sin2(2x+1)+2cos2(2x+1))cos(2x+1)24 \left(- 7 \sin^{2}{\left(2 x + 1 \right)} + 2 \cos^{2}{\left(2 x + 1 \right)}\right) \cos{\left(2 x + 1 \right)}
The graph
Derivative of sin^3(2x+1)