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Derivative of:
Derivative of cosx/(1+sinx^2)
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Similar expressions
cosx/(1-sinx^2)

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

Derivative of cosx/(1+sinx^2)

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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 $\sin^{2}{\left(x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = \sin{\left(x \right)}$ .
  3. Apply the power rule: $u^{2}$ goes to $2 u$
  4. 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:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  The result is: $2 \sin{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

3-th 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)|                 8*cos (x)*|1 - ----------- + ----------- - -----------------||       
|    6*|sin (x) - cos (x) + -----------------|                           |           2             2                     2 ||       
|      |                              2      |         2                 |    1 + sin (x)   1 + sin (x)     /       2   \  ||       
|      \                       1 + sin (x)   /    6*cos (x)              \                                  \1 + sin (x)/  /|       
|1 - ----------------------------------------- + ----------- + -------------------------------------------------------------|*sin(x)
|                          2                            2                                      2                            |       
\                   1 + sin (x)                  1 + sin (x)                            1 + sin (x)                         /       
------------------------------------------------------------------------------------------------------------------------------------
                                                                   2                                                                
                                                            1 + sin (x)

$$\frac{\left(\frac{8 \cdot \left(- \frac{6 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} - \frac{3 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + 1\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + 1 - \frac{6 \left(\frac{4 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}$$

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)|                 8*cos (x)*|1 - ----------- + ----------- - -----------------||       
|    6*|sin (x) - cos (x) + -----------------|                           |           2             2                     2 ||       
|      |                              2      |         2                 |    1 + sin (x)   1 + sin (x)     /       2   \  ||       
|      \                       1 + sin (x)   /    6*cos (x)              \                                  \1 + sin (x)/  /|       
|1 - ----------------------------------------- + ----------- + -------------------------------------------------------------|*sin(x)
|                          2                            2                                      2                            |       
\                   1 + sin (x)                  1 + sin (x)                            1 + sin (x)                         /       
------------------------------------------------------------------------------------------------------------------------------------
                                                                   2                                                                
                                                            1 + sin (x)

Simplify

The graph

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Derivative of cosx/(1+sinx^2)

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The solution