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Identical expressions
y=(sin^ four)x/ four
y equally ( sinus of to the power of 4)x divide by 4
y equally ( sinus of to the power of four)x divide by four
y=(sin4)x/4
y=sin4x/4
y=(sin⁴)x/4
y=sin^4x/4
y=(sin^4)x divide by 4
Similar expressions
y=4sin^4*(x/4)
y=sin^4(x/4)

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

Derivative of y=(sin^4)x/4

The solution

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   4     
sin (x)*x
---------
    4

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

  /   4     \
d |sin (x)*x|
--|---------|
dx\    4    /

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

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \sin^{4}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}$
  The result is: $4 x \sin^{3}{\left(x \right)} \cos{\left(x \right)} + \sin^{4}{\left(x \right)}$
So, the result is: $x \sin^{3}{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{4}{\left(x \right)}}{4}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

   4                      
sin (x)        3          
------- + x*sin (x)*cos(x)
   4

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

Simplify

The second derivative [src]

   2    /    /   2           2   \                  \
sin (x)*\- x*\sin (x) - 3*cos (x)/ + 2*cos(x)*sin(x)/

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

Simplify

The third derivative [src]

 /  /   2           2   \              /       2           2   \       \       
-\3*\sin (x) - 3*cos (x)/*sin(x) + 2*x*\- 3*cos (x) + 5*sin (x)/*cos(x)/*sin(x)

$$- \left(2 x \left(5 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)} + 3 \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}\right) \sin{\left(x \right)}$$

Simplify

The graph

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Derivative of y=(sin^4)x/4

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The solution