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Integral of d{x}:
sin(x/2)*cos(x/2)
Identical expressions
sin(x/ two)*cos(x/ two)
sinus of (x divide by 2) multiply by co sinus of e of (x divide by 2)
sinus of (x divide by two) multiply by co sinus of e of (x divide by two)
sin(x/2)cos(x/2)
sinx/2cosx/2
sin(x divide by 2)*cos(x divide by 2)

The derivative of the function/ sin(x/2)*cos(x/2)

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

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

Apply the product rule:

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

$f{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \frac{x}{2}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $\frac{1}{2}$
  The result of the chain rule is:
  
  $\frac{\cos{\left(\frac{x}{2} \right)}}{2}$
$g{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{x}{2}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $\frac{1}{2}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(\frac{x}{2} \right)}}{2}$
The result is: $- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2/x\      2/x\
cos |-|   sin |-|
    \2/       \2/
------- - -------
   2         2

$$- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}$$

Simplify

The second derivative [src]

    /x\    /x\
-cos|-|*sin|-|
    \2/    \2/

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

Simplify

The third derivative [src]

   2/x\      2/x\
sin |-| - cos |-|
    \2/       \2/
-----------------
        2

$$\frac{\sin^{2}{\left(\frac{x}{2} \right)} - \cos^{2}{\left(\frac{x}{2} \right)}}{2}$$

Simplify

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

The graph:

Piecewise:

The solution