Derivative of y=sin⁹(x/2)

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   9/x\
sin |-|
    \2/

$$\sin^{9}{\left(\frac{x}{2} \right)}$$

sin(x/2)^9

Detail solution

Let $u = \sin{\left(\frac{x}{2} \right)}$ .
Apply the power rule: $u^{9}$ goes to $9 u^{8}$
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:

$\frac{9 \sin^{8}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{2}$
Now simplify:

$\frac{9 \sin^{8}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{2}$

The answer is:

$\frac{9 \sin^{8}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{2}$

The graph

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The first derivative [src]

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

$$\frac{9 \sin^{8}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{2}$$

Simplify

The second derivative [src]

          /               2/x\\
          |            sin |-||
     7/x\ |     2/x\       \2/|
9*sin |-|*|2*cos |-| - -------|
      \2/ \      \2/      4   /

$$9 \left(- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{4} + 2 \cos^{2}{\left(\frac{x}{2} \right)}\right) \sin^{7}{\left(\frac{x}{2} \right)}$$

Simplify

The third derivative [src]

          /                  2/x\\       
          |            25*sin |-||       
     6/x\ |     2/x\          \2/|    /x\
9*sin |-|*|7*cos |-| - ----------|*cos|-|
      \2/ \      \2/       8     /    \2/

$$9 \left(- \frac{25 \sin^{2}{\left(\frac{x}{2} \right)}}{8} + 7 \cos^{2}{\left(\frac{x}{2} \right)}\right) \sin^{6}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}$$

Simplify

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Derivative of y=sin⁹(x/2)

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