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

The solution

You have entered [src]

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

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

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

Detail solution

Let $u = \frac{1 - \sin{\left(x \right)}}{\sin{\left(x \right)} + 1}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1 - \sin{\left(x \right)}}{\sin{\left(x \right)} + 1}$ :
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.
      1. 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:

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

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

The answer is:

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

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

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

Simplify

The second derivative [src]

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

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

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

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

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution