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Identical expressions
sin^ three (x/(2cos(4x)))
sinus of cubed (x divide by (2 co sinus of e of (4x)))
sinus of to the power of three (x divide by (2 co sinus of e of (4x)))
sin3(x/(2cos(4x)))
sin3x/2cos4x
sin³(x/(2cos(4x)))
sin to the power of 3(x/(2cos(4x)))
sin^3x/2cos4x
sin^3(x divide by (2cos(4x)))

The derivative of the function/ sin^3(x/(2cos(4x)))

Derivative of sin^3(x/(2cos(4x)))

The solution

You have entered [src]

   3/    x     \
sin |----------|
    \2*cos(4*x)/

$$\sin^{3}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}$$

sin(x/((2*cos(4*x))))^3

Detail solution

Let $u = \sin{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}$ :
1. Let $u = \frac{x}{2 \cos{\left(4 x \right)}}$ .
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 $\frac{d}{d x} \frac{x}{2 \cos{\left(4 x \right)}}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = x$ and $g{\left(x \right)} = 2 \cos{\left(4 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Let $u = 4 x$ .
      2. The derivative of cosine is negative sine:
        
        $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $4$
        
        The result of the chain rule is:
        
        $- 4 \sin{\left(4 x \right)}$
      So, the result is: $- 8 \sin{\left(4 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{8 x \sin{\left(4 x \right)} + 2 \cos{\left(4 x \right)}}{4 \cos^{2}{\left(4 x \right)}}$
  The result of the chain rule is:
  
  $\frac{\left(8 x \sin{\left(4 x \right)} + 2 \cos{\left(4 x \right)}\right) \cos{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{4 \cos^{2}{\left(4 x \right)}}$
The result of the chain rule is:

$\frac{3 \left(8 x \sin{\left(4 x \right)} + 2 \cos{\left(4 x \right)}\right) \sin^{2}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)} \cos{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{4 \cos^{2}{\left(4 x \right)}}$
Now simplify:

$\frac{3 \left(4 x \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) \sin^{2}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)} \cos{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{2 \cos^{2}{\left(4 x \right)}}$

The answer is:

$\frac{3 \left(4 x \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) \sin^{2}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)} \cos{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{2 \cos^{2}{\left(4 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     2/    x     \ /    1        2*x*sin(4*x)\    /    x     \
3*sin |----------|*|---------- + ------------|*cos|----------|
      \2*cos(4*x)/ |2*cos(4*x)       2       |    \2*cos(4*x)/
                   \              cos (4*x)  /

$$3 \left(\frac{2 x \sin{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + \frac{1}{2 \cos{\left(4 x \right)}}\right) \sin^{2}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)} \cos{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}$$

Simplify

The second derivative [src]

  /                  2                                                                                                           2                 \                
  |/    4*x*sin(4*x)\     2/    x     \                                                                        /    4*x*sin(4*x)\     2/    x     \|                
  ||1 + ------------| *cos |----------|     /                        2     \                                   |1 + ------------| *sin |----------||                
  |\      cos(4*x)  /      \2*cos(4*x)/     |      sin(4*x)   4*x*sin (4*x)|    /    x     \    /    x     \   \      cos(4*x)  /      \2*cos(4*x)/|    /    x     \
3*|------------------------------------ + 4*|2*x + -------- + -------------|*cos|----------|*sin|----------| - ------------------------------------|*sin|----------|
  |             2*cos(4*x)                  |      cos(4*x)        2       |    \2*cos(4*x)/    \2*cos(4*x)/                4*cos(4*x)             |    \2*cos(4*x)/
  \                                         \                   cos (4*x)  /                                                                       /                
--------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                              cos(4*x)

$$\frac{3 \left(- \frac{\left(\frac{4 x \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}} + 1\right)^{2} \sin^{2}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{4 \cos{\left(4 x \right)}} + \frac{\left(\frac{4 x \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{2 \cos{\left(4 x \right)}} + 4 \left(\frac{4 x \sin^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + 2 x + \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}\right) \sin{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)} \cos{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}\right) \sin{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{\cos{\left(4 x \right)}}$$

Simplify

The third derivative [src]

  /                                                                                                                                                                     /                        2     \                                                                                                   /                        2     \                \
  |                                                                                                          3                         3/    x     \ /    4*x*sin(4*x)\ |      sin(4*x)   4*x*sin (4*x)|                       3                                          2/    x     \ /    4*x*sin(4*x)\ |      sin(4*x)   4*x*sin (4*x)|    /    x     \|
  |                                                                                        /    4*x*sin(4*x)\     3/    x     \   6*sin |----------|*|1 + ------------|*|2*x + -------- + -------------|     /    4*x*sin(4*x)\     2/    x     \    /    x     \   12*cos |----------|*|1 + ------------|*|2*x + -------- + -------------|*sin|----------||
  |                   /         2                                3     \                   |1 + ------------| *cos |----------|         \2*cos(4*x)/ \      cos(4*x)  / |      cos(4*x)        2       |   7*|1 + ------------| *sin |----------|*cos|----------|          \2*cos(4*x)/ \      cos(4*x)  / |      cos(4*x)        2       |    \2*cos(4*x)/|
  |     2/    x     \ |    6*sin (4*x)   20*x*sin(4*x)   24*x*sin (4*x)|    /    x     \   \      cos(4*x)  /      \2*cos(4*x)/                                         \                   cos (4*x)  /     \      cos(4*x)  /      \2*cos(4*x)/    \2*cos(4*x)/                                          \                   cos (4*x)  /                |
3*|8*sin |----------|*|3 + ----------- + ------------- + --------------|*cos|----------| + ------------------------------------ - ---------------------------------------------------------------------- - ------------------------------------------------------ + ---------------------------------------------------------------------------------------|
  |      \2*cos(4*x)/ |        2            cos(4*x)          3        |    \2*cos(4*x)/                    2                                                    cos(4*x)                                                            2                                                                      cos(4*x)                                       |
  \                   \     cos (4*x)                      cos (4*x)   /                               4*cos (4*x)                                                                                                              8*cos (4*x)                                                                                                                /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                          cos(4*x)

$$\frac{3 \left(- \frac{7 \left(\frac{4 x \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}} + 1\right)^{3} \sin^{2}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)} \cos{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{8 \cos^{2}{\left(4 x \right)}} + \frac{\left(\frac{4 x \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}} + 1\right)^{3} \cos^{3}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{4 \cos^{2}{\left(4 x \right)}} - \frac{6 \left(\frac{4 x \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}} + 1\right) \left(\frac{4 x \sin^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + 2 x + \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}\right) \sin^{3}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{\cos{\left(4 x \right)}} + \frac{12 \left(\frac{4 x \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}} + 1\right) \left(\frac{4 x \sin^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + 2 x + \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}\right) \sin{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)} \cos^{2}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}}{\cos{\left(4 x \right)}} + 8 \left(\frac{24 x \sin^{3}{\left(4 x \right)}}{\cos^{3}{\left(4 x \right)}} + \frac{20 x \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}} + \frac{6 \sin^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + 3\right) \sin^{2}{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)} \cos{\left(\frac{x}{2 \cos{\left(4 x \right)}} \right)}\right)}{\cos{\left(4 x \right)}}$$

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Derivative of sin^3(x/(2cos(4x)))

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