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Identical expressions
lnsqrt(one -sinx)/(one +sinx)
ln square root of (1 minus sinus of x) divide by (1 plus sinus of x)
ln square root of (one minus sinus of x) divide by (one plus sinus of x)
ln√(1-sinx)/(1+sinx)
lnsqrt1-sinx/1+sinx
lnsqrt(1-sinx) divide by (1+sinx)
Similar expressions
lnsqrt(1+sinx)/(1+sinx)
lnsqrt(1-sinx)/(1-sinx)

The derivative of the function/ lnsqrt(1-sinx)/(1+sinx)

Derivative of lnsqrt(1-sinx)/(1+sinx)

The solution

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   /  ____________\
log\\/ 1 - sin(x) /
-------------------
     1 + sin(x)

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

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(\sqrt{1 - \sin{\left(x \right)}} \right)}$ and $g{\left(x \right)} = \sin{\left(x \right)} + 1$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/            2                       /                         2     \                                                                                  \       
|       2*cos (x)        3*sin(x)    |      6*sin(x)      6*cos (x)  |    /  ____________\       /     2              \         /     2             \   |       
|-1 + -------------- + -----------   |-1 + ---------- + -------------|*log\\/ 1 - sin(x) /       |  cos (x)           |         |2*cos (x)          |   |       
|                  2   -1 + sin(x)   |     1 + sin(x)               2|                         3*|----------- + sin(x)|       3*|---------- + sin(x)|   |       
|     (-1 + sin(x))                  \                  (1 + sin(x)) /                           \-1 + sin(x)         /         \1 + sin(x)         /   |       
|--------------------------------- - ----------------------------------------------------- + ---------------------------- + ----------------------------|*cos(x)
\         2*(-1 + sin(x))                                  1 + sin(x)                        2*(1 + sin(x))*(-1 + sin(x))   2*(1 + sin(x))*(-1 + sin(x))/       
----------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                           1 + sin(x)

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

Simplify

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Derivative of lnsqrt(1-sinx)/(1+sinx)

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