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  • Derivative of x^2+4 Derivative of x^2+4
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  • Identical expressions

  • sin^ two (x/ two)*ctg(x/ two)
  • sinus of squared (x divide by 2) multiply by ctg(x divide by 2)
  • sinus of to the power of two (x divide by two) multiply by ctg(x divide by two)
  • sin2(x/2)*ctg(x/2)
  • sin2x/2*ctgx/2
  • sin²(x/2)*ctg(x/2)
  • sin to the power of 2(x/2)*ctg(x/2)
  • sin^2(x/2)ctg(x/2)
  • sin2(x/2)ctg(x/2)
  • sin2x/2ctgx/2
  • sin^2x/2ctgx/2
  • sin^2(x divide by 2)*ctg(x divide by 2)
The derivative of the function/ sin^2(x/2)*ctg(x/2)

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

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

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The solution

You have entered [src] 
   2/x\    /x\
sin |-|*cot|-|
    \2/    \2/
sin⁡2(x2)cot⁡(x2)\sin^{2}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)} 
sin(x/2)^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)=sin⁡2(x2)f{\left(x \right)} = \sin^{2}{\left(\frac{x}{2} \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=sin⁡(x2)u = \sin{\left(\frac{x}{2} \right)}.

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

    3. Then, apply the chain rule. Multiply by ddxsin⁡(x2)\frac{d}{d x} \sin{\left(\frac{x}{2} \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}

      The result of the chain rule is:

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

    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:

          sin⁡2(x2)2+cos⁡2(x2)2cos⁡2(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:

        −sin⁡2(x2)2+cos⁡2(x2)2cos⁡2(x2)tan⁡2(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:

        −sin⁡2(x2)2−cos⁡2(x2)2sin⁡2(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: −(sin⁡2(x2)2+cos⁡2(x2)2)sin⁡2(x2)cos⁡2(x2)tan⁡2(x2)+sin⁡(x2)cos⁡(x2)cot⁡(x2)- \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}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}} + \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}

  2. Now simplify:

    sin⁡(3x2)+sin⁡(5x2)4(sin⁡(x2)+sin⁡(3x2))\frac{\sin{\left(\frac{3 x}{2} \right)} + \sin{\left(\frac{5 x}{2} \right)}}{4 \left(\sin{\left(\frac{x}{2} \right)} + \sin{\left(\frac{3 x}{2} \right)}\right)}

The answer is:

sin⁡(3x2)+sin⁡(5x2)4(sin⁡(x2)+sin⁡(3x2))\frac{\sin{\left(\frac{3 x}{2} \right)} + \sin{\left(\frac{5 x}{2} \right)}}{4 \left(\sin{\left(\frac{x}{2} \right)} + \sin{\left(\frac{3 x}{2} \right)}\right)}

The graph
02468-8-6-4-2-10101-1
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        /         2/x\\                       
        |      cot |-||                       
   2/x\ |  1       \2/|      /x\    /x\    /x\
sin |-|*|- - - -------| + cos|-|*cot|-|*sin|-|
    \2/ \  2      2   /      \2/    \2/    \2/
(−cot⁡2(x2)2−12)sin⁡2(x2)+sin⁡(x2)cos⁡(x2)cot⁡(x2)\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}\right) \sin^{2}{\left(\frac{x}{2} \right)} + \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}
Simplify
The second derivative [src] 
  /   2/x\      2/x\\    /x\      2/x\ /       2/x\\    /x\                              
  |sin |-| - cos |-||*cot|-|   sin |-|*|1 + cot |-||*cot|-|                              
  \    \2/       \2//    \2/       \2/ \        \2//    \2/   /       2/x\\    /x\    /x\
- -------------------------- + ---------------------------- - |1 + cot |-||*cos|-|*sin|-|
              2                             2                 \        \2//    \2/    \2/
−(sin⁡2(x2)−cos⁡2(x2))cot⁡(x2)2+(cot⁡2(x2)+1)sin⁡2(x2)cot⁡(x2)2−(cot⁡2(x2)+1)sin⁡(x2)cos⁡(x2)- \frac{\left(\sin^{2}{\left(\frac{x}{2} \right)} - \cos^{2}{\left(\frac{x}{2} \right)}\right) \cot{\left(\frac{x}{2} \right)}}{2} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{2} - \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}
Simplify
The third derivative [src] 
  /       2/x\\ /   2/x\      2/x\\                             2/x\ /       2/x\\ /         2/x\\     /       2/x\\    /x\    /x\    /x\
3*|1 + cot |-||*|sin |-| - cos |-||                          sin |-|*|1 + cot |-||*|1 + 3*cot |-||   3*|1 + cot |-||*cos|-|*cot|-|*sin|-|
  \        \2// \    \2/       \2//      /x\    /x\    /x\       \2/ \        \2// \          \2//     \        \2//    \2/    \2/    \2/
----------------------------------- - cos|-|*cot|-|*sin|-| - ------------------------------------- + ------------------------------------
                 4                       \2/    \2/    \2/                     4                                      2
3(sin⁡2(x2)−cos⁡2(x2))(cot⁡2(x2)+1)4−(cot⁡2(x2)+1)(3cot⁡2(x2)+1)sin⁡2(x2)4+3(cot⁡2(x2)+1)sin⁡(x2)cos⁡(x2)cot⁡(x2)2−sin⁡(x2)cos⁡(x2)cot⁡(x2)\frac{3 \left(\sin^{2}{\left(\frac{x}{2} \right)} - \cos^{2}{\left(\frac{x}{2} \right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}{4} - \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}{2} \right)}}{4} + \frac{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{2} - \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}
Simplify
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