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sin(x/3)^(2)*cot(x/2)

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

Function f() - derivative -N order at the point
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   2/x\    /x\
sin |-|*cot|-|
    \3/    \2/
sin2(x3)cot(x2)\sin^{2}{\left(\frac{x}{3} \right)} \cot{\left(\frac{x}{2} \right)}
sin(x/3)^2*cot(x/2)
Detail solution
  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=sin2(x3)f{\left(x \right)} = \sin^{2}{\left(\frac{x}{3} \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=sin(x3)u = \sin{\left(\frac{x}{3} \right)}.

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

    3. Then, apply the chain rule. Multiply by ddxsin(x3)\frac{d}{d x} \sin{\left(\frac{x}{3} \right)}:

      1. Let u=x3u = \frac{x}{3}.

      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 ddxx3\frac{d}{d x} \frac{x}{3}:

        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: 13\frac{1}{3}

        The result of the chain rule is:

        cos(x3)3\frac{\cos{\left(\frac{x}{3} \right)}}{3}

      The result of the chain rule is:

      2sin(x3)cos(x3)3\frac{2 \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}}{3}

    g(x)=cot(x2)g{\left(x \right)} = \cot{\left(\frac{x}{2} \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. There are multiple ways to do this derivative.

      Method #1

      1. Rewrite the function to be differentiated:

        cot(x2)=1tan(x2)\cot{\left(\frac{x}{2} \right)} = \frac{1}{\tan{\left(\frac{x}{2} \right)}}

      2. Let u=tan(x2)u = \tan{\left(\frac{x}{2} \right)}.

      3. Apply the power rule: 1u\frac{1}{u} goes to 1u2- \frac{1}{u^{2}}

      4. Then, apply the chain rule. Multiply by ddxtan(x2)\frac{d}{d x} \tan{\left(\frac{x}{2} \right)}:

        1. Rewrite the function to be differentiated:

          tan(x2)=sin(x2)cos(x2)\tan{\left(\frac{x}{2} \right)} = \frac{\sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}

        2. Apply the quotient rule, which is:

          ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

          f(x)=sin(x2)f{\left(x \right)} = \sin{\left(\frac{x}{2} \right)} and g(x)=cos(x2)g{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}.

          To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

          1. Let u=x2u = \frac{x}{2}.

          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 ddxx2\frac{d}{d x} \frac{x}{2}:

            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: 12\frac{1}{2}

            The result of the chain rule is:

            cos(x2)2\frac{\cos{\left(\frac{x}{2} \right)}}{2}

          To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

          1. Let u=x2u = \frac{x}{2}.

          2. The derivative of cosine is negative sine:

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

          3. Then, apply the chain rule. Multiply by ddxx2\frac{d}{d x} \frac{x}{2}:

            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: 12\frac{1}{2}

            The result of the chain rule is:

            sin(x2)2- \frac{\sin{\left(\frac{x}{2} \right)}}{2}

          Now plug in to the quotient rule:

          sin2(x2)2+cos2(x2)2cos2(x2)\frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)}}

        The result of the chain rule is:

        sin2(x2)2+cos2(x2)2cos2(x2)tan2(x2)- \frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}

      Method #2

      1. Rewrite the function to be differentiated:

        cot(x2)=cos(x2)sin(x2)\cot{\left(\frac{x}{2} \right)} = \frac{\cos{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} \right)}}

      2. Apply the quotient rule, which is:

        ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

        f(x)=cos(x2)f{\left(x \right)} = \cos{\left(\frac{x}{2} \right)} and g(x)=sin(x2)g{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}.

        To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Let u=x2u = \frac{x}{2}.

        2. The derivative of cosine is negative sine:

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

        3. Then, apply the chain rule. Multiply by ddxx2\frac{d}{d x} \frac{x}{2}:

          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: 12\frac{1}{2}

          The result of the chain rule is:

          sin(x2)2- \frac{\sin{\left(\frac{x}{2} \right)}}{2}

        To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Let u=x2u = \frac{x}{2}.

        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 ddxx2\frac{d}{d x} \frac{x}{2}:

          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: 12\frac{1}{2}

          The result of the chain rule is:

          cos(x2)2\frac{\cos{\left(\frac{x}{2} \right)}}{2}

        Now plug in to the quotient rule:

        sin2(x2)2cos2(x2)2sin2(x2)\frac{- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\sin^{2}{\left(\frac{x}{2} \right)}}

    The result is: (sin2(x2)2+cos2(x2)2)sin2(x3)cos2(x2)tan2(x2)+2sin(x3)cos(x3)cot(x2)3- \frac{\left(\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}\right) \sin^{2}{\left(\frac{x}{3} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}} + \frac{2 \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)} \cot{\left(\frac{x}{2} \right)}}{3}

  2. Now simplify:

    cos(x3)+3cos(2x3)cos(5x3)36(cos(x)+1)tan2(x2)\frac{\cos{\left(\frac{x}{3} \right)} + 3 \cos{\left(\frac{2 x}{3} \right)} - \cos{\left(\frac{5 x}{3} \right)} - 3}{6 \left(\cos{\left(x \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} \right)}}


The answer is:

cos(x3)+3cos(2x3)cos(5x3)36(cos(x)+1)tan2(x2)\frac{\cos{\left(\frac{x}{3} \right)} + 3 \cos{\left(\frac{2 x}{3} \right)} - \cos{\left(\frac{5 x}{3} \right)} - 3}{6 \left(\cos{\left(x \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} \right)}}

The graph
02468-8-6-4-2-1010-250250
The first derivative [src]
        /         2/x\\        /x\    /x\    /x\
        |      cot |-||   2*cos|-|*cot|-|*sin|-|
   2/x\ |  1       \2/|        \3/    \2/    \3/
sin |-|*|- - - -------| + ----------------------
    \3/ \  2      2   /             3           
(cot2(x2)212)sin2(x3)+2sin(x3)cos(x3)cot(x2)3\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}\right) \sin^{2}{\left(\frac{x}{3} \right)} + \frac{2 \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)} \cot{\left(\frac{x}{2} \right)}}{3}
The second derivative [src]
    /   2/x\      2/x\\    /x\      /       2/x\\    /x\    /x\        2/x\ /       2/x\\    /x\
- 4*|sin |-| - cos |-||*cot|-| - 12*|1 + cot |-||*cos|-|*sin|-| + 9*sin |-|*|1 + cot |-||*cot|-|
    \    \3/       \3//    \2/      \        \2//    \3/    \3/         \3/ \        \2//    \2/
------------------------------------------------------------------------------------------------
                                               18                                               
4(sin2(x3)cos2(x3))cot(x2)+9(cot2(x2)+1)sin2(x3)cot(x2)12(cot2(x2)+1)sin(x3)cos(x3)18\frac{- 4 \left(\sin^{2}{\left(\frac{x}{3} \right)} - \cos^{2}{\left(\frac{x}{3} \right)}\right) \cot{\left(\frac{x}{2} \right)} + 9 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin^{2}{\left(\frac{x}{3} \right)} \cot{\left(\frac{x}{2} \right)} - 12 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}}{18}
The third derivative [src]
/       2/x\\ /   2/x\      2/x\\        /x\    /x\    /x\      2/x\ /       2/x\\ /         2/x\\                                     
|1 + cot |-||*|sin |-| - cos |-||   8*cos|-|*cot|-|*sin|-|   sin |-|*|1 + cot |-||*|1 + 3*cot |-||                                     
\        \2// \    \3/       \3//        \3/    \2/    \3/       \3/ \        \2// \          \2//   /       2/x\\    /x\    /x\    /x\
--------------------------------- - ---------------------- - ------------------------------------- + |1 + cot |-||*cos|-|*cot|-|*sin|-|
                3                             27                               4                     \        \2//    \3/    \2/    \3/
(sin2(x3)cos2(x3))(cot2(x2)+1)3(cot2(x2)+1)(3cot2(x2)+1)sin2(x3)4+(cot2(x2)+1)sin(x3)cos(x3)cot(x2)8sin(x3)cos(x3)cot(x2)27\frac{\left(\sin^{2}{\left(\frac{x}{3} \right)} - \cos^{2}{\left(\frac{x}{3} \right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}{3} - \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(3 \cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin^{2}{\left(\frac{x}{3} \right)}}{4} + \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)} \cot{\left(\frac{x}{2} \right)} - \frac{8 \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)} \cot{\left(\frac{x}{2} \right)}}{27}
The graph
Derivative of sin(x/3)^(2)*cot(x/2)