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Derivative of sin3x+2sin^3*(x)

Function f() - derivative -N order at the point
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                3   
sin(3*x) + 2*sin (x)
2sin3(x)+sin(3x)2 \sin^{3}{\left(x \right)} + \sin{\left(3 x \right)}
sin(3*x) + 2*sin(x)^3
Detail solution
  1. Differentiate 2sin3(x)+sin(3x)2 \sin^{3}{\left(x \right)} + \sin{\left(3 x \right)} term by term:

    1. Let u=3xu = 3 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 ddx3x\frac{d}{d x} 3 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: 33

      The result of the chain rule is:

      3cos(3x)3 \cos{\left(3 x \right)}

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

      1. Let u=sin(x)u = \sin{\left(x \right)}.

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

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

        1. The derivative of sine is cosine:

          ddxsin(x)=cos(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        The result of the chain rule is:

        3sin2(x)cos(x)3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}

      So, the result is: 6sin2(x)cos(x)6 \sin^{2}{\left(x \right)} \cos{\left(x \right)}

    The result is: 6sin2(x)cos(x)+3cos(3x)6 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 3 \cos{\left(3 x \right)}

  2. Now simplify:

    3cos(x)2+3cos(3x)2\frac{3 \cos{\left(x \right)}}{2} + \frac{3 \cos{\left(3 x \right)}}{2}


The answer is:

3cos(x)2+3cos(3x)2\frac{3 \cos{\left(x \right)}}{2} + \frac{3 \cos{\left(3 x \right)}}{2}

The graph
02468-8-6-4-2-10105-5
The first derivative [src]
                  2          
3*cos(3*x) + 6*sin (x)*cos(x)
6sin2(x)cos(x)+3cos(3x)6 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 3 \cos{\left(3 x \right)}
The second derivative [src]
  /                   3           2          \
3*\-3*sin(3*x) - 2*sin (x) + 4*cos (x)*sin(x)/
3(2sin3(x)+4sin(x)cos2(x)3sin(3x))3 \left(- 2 \sin^{3}{\left(x \right)} + 4 \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 3 \sin{\left(3 x \right)}\right)
The third derivative [src]
  /                   3            2          \
3*\-9*cos(3*x) + 4*cos (x) - 14*sin (x)*cos(x)/
3(14sin2(x)cos(x)+4cos3(x)9cos(3x))3 \left(- 14 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 4 \cos^{3}{\left(x \right)} - 9 \cos{\left(3 x \right)}\right)