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Derivative of sin(x/2)*cos(x/2)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /x\    /x\
sin|-|*cos|-|
   \2/    \2/
$$\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}$$
sin(x/2)*cos(x/2)
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Let .

    2. The derivative of sine is cosine:

    3. Then, apply the chain rule. Multiply by :

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

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    ; to find :

    1. Let .

    2. The derivative of cosine is negative sine:

    3. Then, apply the chain rule. Multiply by :

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

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
   2/x\      2/x\
cos |-|   sin |-|
    \2/       \2/
------- - -------
   2         2   
$$- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}$$
The second derivative [src]
    /x\    /x\
-cos|-|*sin|-|
    \2/    \2/
$$- \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}$$
The third derivative [src]
   2/x\      2/x\
sin |-| - cos |-|
    \2/       \2/
-----------------
        2        
$$\frac{\sin^{2}{\left(\frac{x}{2} \right)} - \cos^{2}{\left(\frac{x}{2} \right)}}{2}$$