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Identical expressions
log(sqrt((one -sin(x))/(one +sin(x))))
logarithm of ( square root of ((1 minus sinus of (x)) divide by (1 plus sinus of (x))))
logarithm of ( square root of ((one minus sinus of (x)) divide by (one plus sinus of (x))))
log(√((1-sin(x))/(1+sin(x))))
logsqrt1-sinx/1+sinx
log(sqrt((1-sin(x)) divide by (1+sin(x))))
Similar expressions
log(sqrt((1-sinx)/(1+sinx)))
log(sqrt((1+sin(x))/(1+sin(x))))
log(sqrt((1-sin(x))/(1-sin(x))))
log(sqrt((1-sinx)/(1+sinx)))

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

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

The solution

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

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

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

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

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