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Similar expressions
1/(sin^2x-2)

The derivative of the function/ 1/(sin^2x+2)

Derivative of 1/(sin^2x+2)

The solution

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

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

1/(sin(x)^2 + 2)

Detail solution

Let $u = \sin^{2}{\left(x \right)} + 2$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sin^{2}{\left(x \right)} + 2\right)$ :
1. Differentiate $\sin^{2}{\left(x \right)} + 2$ term by term:
  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:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  4. The derivative of the constant $2$ is zero.
  The result is: $2 \sin{\left(x \right)} \cos{\left(x \right)}$
The result of the chain rule is:

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

$- \frac{4 \sin{\left(2 x \right)}}{\left(\cos{\left(2 x \right)} - 5\right)^{2}}$

The answer is:

$- \frac{4 \sin{\left(2 x \right)}}{\left(\cos{\left(2 x \right)} - 5\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of 1/(sin^2x+2)

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The solution