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Identical expressions
y=sin^ two (x/ two)*sinx
y equally sinus of squared (x divide by 2) multiply by sinus of x
y equally sinus of to the power of two (x divide by two) multiply by sinus of x
y=sin2(x/2)*sinx
y=sin2x/2*sinx
y=sin²(x/2)*sinx
y=sin to the power of 2(x/2)*sinx
y=sin^2(x/2)sinx
y=sin2(x/2)sinx
y=sin2x/2sinx
y=sin^2x/2sinx
y=sin^2(x divide by 2)*sinx

The derivative of the function/ y=sin^2(x/2)*sinx

Derivative of y=sin^2(x/2)*sinx

The solution

You have entered [src]

   2/x\       
sin |-|*sin(x)
    \2/

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

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

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2/x\             /x\           /x\
sin |-|*cos(x) + cos|-|*sin(x)*sin|-|
    \2/             \2/           \2/

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

Simplify

The second derivative [src]

                   /   2/x\      2/x\\                                
                   |sin |-| - cos |-||*sin(x)                         
     2/x\          \    \2/       \2//                      /x\    /x\
- sin |-|*sin(x) - -------------------------- + 2*cos(x)*cos|-|*sin|-|
      \2/                      2                            \2/    \2/

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

Simplify

The third derivative [src]

 /                   /   2/x\      2/x\\                                \
 |                 3*|sin |-| - cos |-||*cos(x)                         |
 |   2/x\            \    \2/       \2//               /x\           /x\|
-|sin |-|*cos(x) + ---------------------------- + 4*cos|-|*sin(x)*sin|-||
 \    \2/                       2                      \2/           \2//

$$- (\frac{3 \left(\sin^{2}{\left(\frac{x}{2} \right)} - \cos^{2}{\left(\frac{x}{2} \right)}\right) \cos{\left(x \right)}}{2} + \sin^{2}{\left(\frac{x}{2} \right)} \cos{\left(x \right)} + 4 \sin{\left(\frac{x}{2} \right)} \sin{\left(x \right)} \cos{\left(\frac{x}{2} \right)})$$

Simplify

The graph

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Identical expressions

Derivative of y=sin^2(x/2)*sinx

The graph:

Piecewise:

The solution