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Derivative of:
Derivative of 8^x
Derivative of 2*sinh(x)*cosh(x)
Derivative of x^2*asinh(x)*acosh(x)
Derivative of (x^2-4)/x
Integral of d{x}:
sin^8(x/4)
Identical expressions
sin^ eight (x/ four)
sinus of to the power of 8(x divide by 4)
sinus of to the power of eight (x divide by four)
sin8(x/4)
sin8x/4
sin⁸(x/4)
sin^8x/4
sin^8(x divide by 4)

The derivative of the function/ sin^8(x/4)

Derivative of sin^8(x/4)

The solution

You have entered [src]

   8/x\
sin |-|
    \4/

$$\sin^{8}{\left(\frac{x}{4} \right)}$$

sin(x/4)^8

Detail solution

Let $u = \sin{\left(\frac{x}{4} \right)}$ .
Apply the power rule: $u^{8}$ goes to $8 u^{7}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\frac{x}{4} \right)}$ :
1. Let $u = \frac{x}{4}$ .
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 $\frac{d}{d x} \frac{x}{4}$ :
  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: $\frac{1}{4}$
  The result of the chain rule is:
  
  $\frac{\cos{\left(\frac{x}{4} \right)}}{4}$
The result of the chain rule is:

$2 \sin^{7}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   6/x\ /     2/x\        2/x\\
sin |-|*|- sin |-| + 7*cos |-||
    \4/ \      \4/         \4//
-------------------------------
               2

$$\frac{\left(- \sin^{2}{\left(\frac{x}{4} \right)} + 7 \cos^{2}{\left(\frac{x}{4} \right)}\right) \sin^{6}{\left(\frac{x}{4} \right)}}{2}$$

Simplify

The third derivative [src]

   5/x\ /        2/x\         2/x\\    /x\
sin |-|*|- 11*sin |-| + 21*cos |-||*cos|-|
    \4/ \         \4/          \4//    \4/
------------------------------------------
                    4

$$\frac{\left(- 11 \sin^{2}{\left(\frac{x}{4} \right)} + 21 \cos^{2}{\left(\frac{x}{4} \right)}\right) \sin^{5}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{4}$$

Simplify

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Identical expressions

Derivative of sin^8(x/4)

The graph:

Piecewise:

The solution