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The derivative of the function/ (3x-sin(3x))/(tan(2x))^3

Derivative of (3x-sin(3x))/(tan(2x))^3

The solution

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3*x - sin(3*x)
--------------
     3        
  tan (2*x)

$$\frac{3 x - \sin{\left(3 x \right)}}{\tan^{3}{\left(2 x \right)}}$$

(3*x - sin(3*x))/tan(2*x)^3

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 3 x - \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \tan^{3}{\left(2 x \right)}$ .

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

$\frac{\left(3 - 3 \cos{\left(3 x \right)}\right) \tan^{3}{\left(2 x \right)} - \frac{3 \left(3 x - \sin{\left(3 x \right)}\right) \left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \tan^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}}{\tan^{6}{\left(2 x \right)}}$
Now simplify:

$\frac{3 \left(- 6 x - \frac{\sin{\left(x \right)}}{4} + 2 \sin{\left(3 x \right)} + \frac{\sin{\left(4 x \right)}}{2} - \frac{\sin{\left(7 x \right)}}{4}\right)}{\cos^{2}{\left(2 x \right)} \tan^{4}{\left(2 x \right)}}$

The answer is:

$\frac{3 \left(- 6 x - \frac{\sin{\left(x \right)}}{4} + 2 \sin{\left(3 x \right)} + \frac{\sin{\left(4 x \right)}}{2} - \frac{\sin{\left(7 x \right)}}{4}\right)}{\cos^{2}{\left(2 x \right)} \tan^{4}{\left(2 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 /         2     \                 
3 - 3*cos(3*x)   \6 + 6*tan (2*x)/*(3*x - sin(3*x))
-------------- - ----------------------------------
     3                          4                  
  tan (2*x)                  tan (2*x)

$$\frac{3 - 3 \cos{\left(3 x \right)}}{\tan^{3}{\left(2 x \right)}} - \frac{\left(3 x - \sin{\left(3 x \right)}\right) \left(6 \tan^{2}{\left(2 x \right)} + 6\right)}{\tan^{4}{\left(2 x \right)}}$$

Simplify

The second derivative [src]

  /                               /       /       2     \\                        /       2     \                \
  |               /       2     \ |     2*\1 + tan (2*x)/|                     12*\1 + tan (2*x)/*(-1 + cos(3*x))|
3*|3*sin(3*x) + 8*\1 + tan (2*x)/*|-1 + -----------------|*(-sin(3*x) + 3*x) + ----------------------------------|
  |                               |            2         |                                  tan(2*x)             |
  \                               \         tan (2*x)    /                                                       /
------------------------------------------------------------------------------------------------------------------
                                                       3                                                          
                                                    tan (2*x)

$$\frac{3 \left(8 \left(3 x - \sin{\left(3 x \right)}\right) \left(\frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{\tan^{2}{\left(2 x \right)}} - 1\right) \left(\tan^{2}{\left(2 x \right)} + 1\right) + \frac{12 \left(\cos{\left(3 x \right)} - 1\right) \left(\tan^{2}{\left(2 x \right)} + 1\right)}{\tan{\left(2 x \right)}} + 3 \sin{\left(3 x \right)}\right)}{\tan^{3}{\left(2 x \right)}}$$

Simplify

The third derivative [src]

  /                                                                                                                                                    /       /       2     \\                \
  |                                                                                                                                    /       2     \ |     2*\1 + tan (2*x)/|                |
  |                                                                                /                                           2\   72*\1 + tan (2*x)/*|-1 + -----------------|*(-1 + cos(3*x))|
  |                /       2     \                                                 |       /       2     \      /       2     \ |                      |            2         |                |
  |9*cos(3*x)   54*\1 + tan (2*x)/*sin(3*x)      /       2     \                   |    11*\1 + tan (2*x)/   10*\1 + tan (2*x)/ |                      \         tan (2*x)    /                |
3*|---------- - --------------------------- - 16*\1 + tan (2*x)/*(-sin(3*x) + 3*x)*|2 - ------------------ + -------------------| - -----------------------------------------------------------|
  | tan(2*x)                2                                                      |           2                     4          |                             tan(2*x)                         |
  \                      tan (2*x)                                                 \        tan (2*x)             tan (2*x)     /                                                              /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                              2                                                                                                 
                                                                                           tan (2*x)

$$\frac{3 \left(- 16 \left(3 x - \sin{\left(3 x \right)}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\frac{10 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2}}{\tan^{4}{\left(2 x \right)}} - \frac{11 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{\tan^{2}{\left(2 x \right)}} + 2\right) - \frac{72 \left(\frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{\tan^{2}{\left(2 x \right)}} - 1\right) \left(\cos{\left(3 x \right)} - 1\right) \left(\tan^{2}{\left(2 x \right)} + 1\right)}{\tan{\left(2 x \right)}} - \frac{54 \left(\tan^{2}{\left(2 x \right)} + 1\right) \sin{\left(3 x \right)}}{\tan^{2}{\left(2 x \right)}} + \frac{9 \cos{\left(3 x \right)}}{\tan{\left(2 x \right)}}\right)}{\tan^{2}{\left(2 x \right)}}$$

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Derivative of (3x-sin(3x))/(tan(2x))^3

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The solution