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Derivative of:
Derivative of (2*x-1)^2
Derivative of 1/(x+2)
Derivative of (2*e)^cos(log(6*x-1))^(2)
Derivative of 1/tg(x)
Identical expressions
ln√((one +sinx)/(one -sinx))
ln√((1 plus sinus of x) divide by (1 minus sinus of x))
ln√((one plus sinus of x) divide by (one minus sinus of x))
ln√1+sinx/1-sinx
ln√((1+sinx) divide by (1-sinx))
Similar expressions
ln√((1-sinx)/(1-sinx))
ln√((1+sinx)/(1+sinx))

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

Derivative of ln√((1+sinx)/(1-sinx))

The solution

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

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

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

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

Transform

Detail solution

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

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

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

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)/   (1 + sin(x))*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)

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

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

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

Simplify

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Derivative of ln√((1+sinx)/(1-sinx))

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