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sin^9(x/2)

Derivative of sin^9(x/2)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   9/x\
sin |-|
    \2/
$$\sin^{9}{\left(\frac{x}{2} \right)}$$
d /   9/x\\
--|sin |-||
dx\    \2//
$$\frac{d}{d x} \sin^{9}{\left(\frac{x}{2} \right)}$$
Detail solution
  1. Let .

  2. Apply the power rule: goes to

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

    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:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
     8/x\    /x\
9*sin |-|*cos|-|
      \2/    \2/
----------------
       2        
$$\frac{9 \sin^{8}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{2}$$
The second derivative [src]
          /               2/x\\
          |            sin |-||
     7/x\ |     2/x\       \2/|
9*sin |-|*|2*cos |-| - -------|
      \2/ \      \2/      4   /
$$9 \left(- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{4} + 2 \cos^{2}{\left(\frac{x}{2} \right)}\right) \sin^{7}{\left(\frac{x}{2} \right)}$$
The third derivative [src]
          /                  2/x\\       
          |            25*sin |-||       
     6/x\ |     2/x\          \2/|    /x\
9*sin |-|*|7*cos |-| - ----------|*cos|-|
      \2/ \      \2/       8     /    \2/
$$9 \left(- \frac{25 \sin^{2}{\left(\frac{x}{2} \right)}}{8} + 7 \cos^{2}{\left(\frac{x}{2} \right)}\right) \sin^{6}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}$$
The graph
Derivative of sin^9(x/2)