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sin^2(x/2)

Derivative of sin^2(x/2)

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

You have entered [src]
   2/x\
sin |-|
    \2/
sin2(x2)\sin^{2}{\left(\frac{x}{2} \right)}
sin(x/2)^2
Detail solution
  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)}

  4. Now simplify:

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


The answer is:

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

The graph
02468-8-6-4-2-10102-2
The first derivative [src]
   /x\    /x\
cos|-|*sin|-|
   \2/    \2/
sin(x2)cos(x2)\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}
The second derivative [src]
   2/x\      2/x\
cos |-| - sin |-|
    \2/       \2/
-----------------
        2        
sin2(x2)+cos2(x2)2\frac{- \sin^{2}{\left(\frac{x}{2} \right)} + \cos^{2}{\left(\frac{x}{2} \right)}}{2}
The third derivative [src]
    /x\    /x\
-cos|-|*sin|-|
    \2/    \2/
sin(x2)cos(x2)- \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}
The graph
Derivative of sin^2(x/2)