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Derivative of 1/(2+cos(x))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
    1     
----------
2 + cos(x)
$$\frac{1}{\cos{\left(x \right)} + 2}$$
1/(2 + cos(x))
Detail solution
  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. The derivative of cosine is negative sine:

      The result is:

    The result of the chain rule is:


The answer is:

The graph
The first derivative [src]
    sin(x)   
-------------
            2
(2 + cos(x)) 
$$\frac{\sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 2\right)^{2}}$$
The second derivative [src]
     2             
2*sin (x)          
---------- + cos(x)
2 + cos(x)         
-------------------
               2   
   (2 + cos(x))    
$$\frac{\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 2}}{\left(\cos{\left(x \right)} + 2\right)^{2}}$$
The third derivative [src]
/                         2     \       
|      6*cos(x)      6*sin (x)  |       
|-1 + ---------- + -------------|*sin(x)
|     2 + cos(x)               2|       
\                  (2 + cos(x)) /       
----------------------------------------
                         2              
             (2 + cos(x))               
$$\frac{\left(-1 + \frac{6 \cos{\left(x \right)}}{\cos{\left(x \right)} + 2} + \frac{6 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 2\right)^{2}}\right) \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 2\right)^{2}}$$