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Identical expressions
(one +sin(2x))/(one -sin(2x))+tg(2x)
(1 plus sinus of (2x)) divide by (1 minus sinus of (2x)) plus tg(2x)
(one plus sinus of (2x)) divide by (one minus sinus of (2x)) plus tg(2x)
1+sin2x/1-sin2x+tg2x
(1+sin(2x)) divide by (1-sin(2x))+tg(2x)
Similar expressions
(1-sin(2x))/(1-sin(2x))+tg(2x)
(1+sin(2x))/(1-sin(2x))-tg(2x)
(1+sin(2x))/(1+sin(2x))+tg(2x)

The derivative of the function/ (1+sin(2x))/(1-sin(2x))+tg(2x)

Derivative of (1+sin(2x))/(1-sin(2x))+tg(2x)

The solution

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1 + sin(2*x)           
------------ + tan(2*x)
1 - sin(2*x)

$$\tan{\left(2 x \right)} + \frac{\sin{\left(2 x \right)} + 1}{1 - \sin{\left(2 x \right)}}$$

(1 + sin(2*x))/(1 - sin(2*x)) + tan(2*x)

Detail solution

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

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

         2         2*cos(2*x)    2*(1 + sin(2*x))*cos(2*x)
2 + 2*tan (2*x) + ------------ + -------------------------
                  1 - sin(2*x)                      2     
                                      (1 - sin(2*x))

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                 2                          3                                                                                              3                                                         \
  |  /       2     \       cos(2*x)       6*cos (2*x)           2      /       2     \   (1 + sin(2*x))*cos(2*x)   6*cos(2*x)*sin(2*x)   6*cos (2*x)*(1 + sin(2*x))   6*(1 + sin(2*x))*cos(2*x)*sin(2*x)|
8*|2*\1 + tan (2*x)/  + ------------- - ---------------- + 4*tan (2*x)*\1 + tan (2*x)/ - ----------------------- - ------------------- + -------------------------- + ----------------------------------|
  |                     -1 + sin(2*x)                  3                                                    2                       2                        4                                3         |
  \                                     (-1 + sin(2*x))                                      (-1 + sin(2*x))         (-1 + sin(2*x))          (-1 + sin(2*x))                  (-1 + sin(2*x))          /

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

Simplify

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Derivative of (1+sin(2x))/(1-sin(2x))+tg(2x)

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The solution