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Identical expressions
(cos^2x)/(one -sen^2x)
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( co sinus of e of squared x) divide by (one minus sen squared x)
(cos2x)/(1-sen2x)
cos2x/1-sen2x
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Similar expressions
(cos^2x)/(1+sen^2x)

The derivative of the function/ (cos^2x)/(1-sen^2x)

Derivative of (cos^2x)/(1-sen^2x)

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cos^{2}{\left(x \right)}$ and $g{\left(x \right)} = 1 - \sin^{2}{\left(x \right)}$ .

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

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

$0$

The answer is:

$0$

The first derivative [src]

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

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

Simplify

The second derivative [src]

   /                            /                         2       2   \                    \
   |                       2    |   2         2      4*cos (x)*sin (x)|                    |
   |                    cos (x)*|sin (x) - cos (x) + -----------------|                    |
   |                            |                               2     |        2       2   |
   |   2         2              \                       -1 + sin (x)  /   4*cos (x)*sin (x)|
-2*|sin (x) - cos (x) + ----------------------------------------------- + -----------------|
   |                                              2                                  2     |
   \                                      -1 + sin (x)                       -1 + sin (x)  /
--------------------------------------------------------------------------------------------
                                                2                                           
                                        -1 + sin (x)

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (cos^2x)/(1-sen^2x)

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