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Derivative of sin^8(x/4)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   8/x\
sin |-|
    \4/
$$\sin^{8}{\left(\frac{x}{4} \right)}$$
sin(x/4)^8
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]
     7/x\    /x\
2*sin |-|*cos|-|
      \4/    \4/
$$2 \sin^{7}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}$$
The second derivative [src]
   6/x\ /     2/x\        2/x\\
sin |-|*|- sin |-| + 7*cos |-||
    \4/ \      \4/         \4//
-------------------------------
               2               
$$\frac{\left(- \sin^{2}{\left(\frac{x}{4} \right)} + 7 \cos^{2}{\left(\frac{x}{4} \right)}\right) \sin^{6}{\left(\frac{x}{4} \right)}}{2}$$
The third derivative [src]
   5/x\ /        2/x\         2/x\\    /x\
sin |-|*|- 11*sin |-| + 21*cos |-||*cos|-|
    \4/ \         \4/          \4//    \4/
------------------------------------------
                    4                     
$$\frac{\left(- 11 \sin^{2}{\left(\frac{x}{4} \right)} + 21 \cos^{2}{\left(\frac{x}{4} \right)}\right) \sin^{5}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{4}$$