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Identical expressions
ln(sqrt(one -sin(2x)/(one +sin(2x))))
ln( square root of (1 minus sinus of (2x) divide by (1 plus sinus of (2x))))
ln( square root of (one minus sinus of (2x) divide by (one plus sinus of (2x))))
ln(√(1-sin(2x)/(1+sin(2x))))
lnsqrt1-sin2x/1+sin2x
ln(sqrt(1-sin(2x) divide by (1+sin(2x))))
Similar expressions
ln(sqrt(1-sin(2x)/(1-sin(2x))))
ln(sqrt(1+sin(2x)/(1+sin(2x))))

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

Derivative of ln(sqrt(1-sin(2x)/(1+sin(2x))))

The solution

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   /    __________________\
   |   /       sin(2*x)   |
log|  /  1 - ------------ |
   \\/       1 + sin(2*x) /

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

log(sqrt(1 - sin(2*x)/(1 + sin(2*x))))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(sqrt(1-sin(2x)/(1+sin(2x))))

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The solution