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Integral of d{x}:
sin^3(x/2)
Identical expressions
y=sin^ three (x/ two)
y equally sinus of cubed (x divide by 2)
y equally sinus of to the power of three (x divide by two)
y=sin3(x/2)
y=sin3x/2
y=sin³(x/2)
y=sin to the power of 3(x/2)
y=sin^3x/2
y=sin^3(x divide by 2)
Similar expressions
y=sin^3x/2

The derivative of the function/ y=sin^3(x/2)

Derivative of y=sin^3(x/2)

The solution

You have entered [src]

   3/x\
sin |-|
    \2/

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

sin(x/2)^3

Detail solution

Let $u = \sin{\left(\frac{x}{2} \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
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:

$\frac{3 \sin^{2}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{2}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       2/x\        2/x\\    /x\
3*|- 7*sin |-| + 2*cos |-||*cos|-|
  \        \2/         \2//    \2/
----------------------------------
                8

$$\frac{3 \left(- 7 \sin^{2}{\left(\frac{x}{2} \right)} + 2 \cos^{2}{\left(\frac{x}{2} \right)}\right) \cos{\left(\frac{x}{2} \right)}}{8}$$

Simplify

The graph

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Derivative of y=sin^3(x/2)

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The solution