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(three *tan(two *x)/sin(x)/ four)
(3 multiply by tangent of (2 multiply by x) divide by sinus of (x) divide by 4)
(three multiply by tangent of (two multiply by x) divide by sinus of (x) divide by four)
(3tan(2x)/sin(x)/4)
3tan2x/sinx/4
(3*tan(2*x) divide by sin(x) divide by 4)
Similar expressions
(3*tan(2*x)/sinx/4)

The derivative of the function/ (3*tan(2*x)/sin(x)/4)

Derivative of (3tan(2x)/sin(x)/4)

The solution

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

$$\frac{\frac{1}{\sin{\left(x \right)}} 3 \tan{\left(2 x \right)}}{4}$$

((3*tan(2*x))/sin(x))/4

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \tan{\left(2 x \right)}$ and $g{\left(x \right)} = \sin{\left(x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    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 $\frac{d}{d x} 2 x$ :
        
        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 $\frac{d}{d x} 2 x$ :
        
        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:
      
      $\frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{\frac{\left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(2 x \right)}} - \cos{\left(x \right)} \tan{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}$
  So, the result is: $\frac{3 \left(\frac{\left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(2 x \right)}} - \cos{\left(x \right)} \tan{\left(2 x \right)}\right)}{\sin^{2}{\left(x \right)}}$
So, the result is: $\frac{3 \left(\frac{\left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(2 x \right)}} - \cos{\left(x \right)} \tan{\left(2 x \right)}\right)}{4 \sin^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

  //         2   \                                           /       2     \       \
  ||    2*cos (x)|              /       2     \            4*\1 + tan (2*x)/*cos(x)|
3*||1 + ---------|*tan(2*x) + 8*\1 + tan (2*x)/*tan(2*x) - ------------------------|
  ||        2    |                                                  sin(x)         |
  \\     sin (x) /                                                                 /
------------------------------------------------------------------------------------
                                      4*sin(x)

$$\frac{3 \left(\left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \tan{\left(2 x \right)} + 8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} - \frac{4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)}{4 \sin{\left(x \right)}}$$

Simplify

The third derivative [src]

  /                                                                           /         2   \                                                     \
  |                                                                           |    6*cos (x)|                                                     |
  |                                                                           |5 + ---------|*cos(x)*tan(2*x)                                     |
  |                  /         2   \                                          |        2    |                      /       2     \                |
  |  /       2     \ |    2*cos (x)|      /       2     \ /         2     \   \     sin (x) /                   24*\1 + tan (2*x)/*cos(x)*tan(2*x)|
3*|6*\1 + tan (2*x)/*|1 + ---------| + 16*\1 + tan (2*x)/*\1 + 3*tan (2*x)/ - ------------------------------- - ----------------------------------|
  |                  |        2    |                                                       sin(x)                             sin(x)              |
  \                  \     sin (x) /                                                                                                              /
---------------------------------------------------------------------------------------------------------------------------------------------------
                                                                      4*sin(x)

$$\frac{3 \left(6 \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right) - \frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)} \tan{\left(2 x \right)}}{\sin{\left(x \right)}} + 16 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(3 \tan^{2}{\left(2 x \right)} + 1\right) - \frac{24 \left(\tan^{2}{\left(2 x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(2 x \right)}}{\sin{\left(x \right)}}\right)}{4 \sin{\left(x \right)}}$$

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Derivative of (3tan(2x)/sin(x)/4)

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Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (3*tan(2*x)/sin(x)/4)

The graph:

Piecewise:

The solution

Derivative of (3tan(2x)/sin(x)/4)