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Derivative of:
Derivative of x^x^x
Derivative of x^-11
Derivative of 2x²
Derivative of (x^2+1)^sqrt(x)
Integral of d{x}:
1/(2+cos(x))
Limit of the function:
1/(2+cos(x))
Identical expressions
one /(two +cos(x))
1 divide by (2 plus co sinus of e of (x))
one divide by (two plus co sinus of e of (x))
1/2+cosx
1 divide by (2+cos(x))
Similar expressions
1/(2-cos(x))
1/(2+cosx)

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

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

The solution

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

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

1/(2 + cos(x))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

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))

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

Simplify

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

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The solution