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Derivative of 2x²
Derivative of x^2*sqrt(1-x^2)
Derivative of y=3x²+5x+4
Derivative of x^3*cot(x)
Identical expressions
(cosx/(one -sinx))^ two
( co sinus of e of x divide by (1 minus sinus of x)) squared
( co sinus of e of x divide by (one minus sinus of x)) to the power of two
(cosx/(1-sinx))2
cosx/1-sinx2
(cosx/(1-sinx))²
(cosx/(1-sinx)) to the power of 2
cosx/1-sinx^2
(cosx divide by (1-sinx))^2
Similar expressions
(cosx/(1+sinx))^2

The derivative of the function/ (cosx/(1-sinx))^2

Derivative of (cosx/(1-sinx))^2

The solution

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

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

(cos(x)/(1 - sin(x)))^2

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                                                                                                        /     2              \ /                           4              2          \                                                                                                 2                                                                     2                                                \
  |                                                                                                                        |  cos (x)           | |   2         2        3*cos (x)      5*cos (x)*sin(x)|                                                                           /     2              \                     /     2              \     /     2              \             /     2              \              |
  |                                                                                                                      2*|----------- + sin(x)|*|sin (x) - cos (x) + -------------- + ----------------|                                                                           |  cos (x)           |                3    |  cos (x)           |     |  cos (x)           |             |  cos (x)           |              |
  |  /     2              \          /                2             2              4               2          \            \-1 + sin(x)         / |                                 2     -1 + sin(x)   |     /     2              \ /            2                    \          2*|----------- + sin(x)| *cos(x)   2*cos (x)*|----------- + sin(x)|   2*|----------- + sin(x)| *sin(x)   3*|----------- + sin(x)|*cos(x)*sin(x)|
  |  |  cos (x)           |          |           4*cos (x)     3*sin (x)      6*cos (x)      12*cos (x)*sin(x)|                                   \                    (-1 + sin(x))                    /     |  cos (x)           | |       2*cos (x)        3*sin(x) |            \-1 + sin(x)         /                     \-1 + sin(x)         /     \-1 + sin(x)         /             \-1 + sin(x)         /              |
2*|- |----------- + sin(x)|*cos(x) - |-sin(x) - ----------- + ----------- + -------------- + -----------------|*cos(x) - -------------------------------------------------------------------------------- - 2*|----------- + sin(x)|*|-1 + -------------- + -----------|*cos(x) + -------------------------------- + -------------------------------- + -------------------------------- + --------------------------------------|
  |  \-1 + sin(x)         /          |          -1 + sin(x)   -1 + sin(x)                3                  2 |                                               cos(x)                                          \-1 + sin(x)         / |                  2   -1 + sin(x)|                    -1 + sin(x)                                    2                         cos(x)                             -1 + sin(x)              |
  \                                  \                                      (-1 + sin(x))      (-1 + sin(x))  /                                                                                                                      \     (-1 + sin(x))               /                                                      (-1 + sin(x))                                                                                      /
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                       2                                                                                                                                                                                                          
                                                                                                                                                                                                          (-1 + sin(x))

$$\frac{2 \left(\frac{2 \left(\sin{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right)^{2} \sin{\left(x \right)}}{\cos{\left(x \right)}} - 2 \left(\sin{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \left(-1 + \frac{3 \sin{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{2 \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}\right) \cos{\left(x \right)} - \frac{2 \left(\sin{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{5 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{3 \cos^{4}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}\right)}{\cos{\left(x \right)}} - \left(\sin{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \cos{\left(x \right)} - \left(- \sin{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1} - \frac{4 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{12 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}} + \frac{6 \cos^{4}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{3}}\right) \cos{\left(x \right)} + \frac{2 \left(\sin{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right)^{2} \cos{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{3 \left(\sin{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{2 \left(\sin{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \cos^{3}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}\right)}{\left(\sin{\left(x \right)} - 1\right)^{2}}$$

Simplify

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Derivative of (cosx/(1-sinx))^2

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The solution