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The derivative of the function/ sqrt(1+cos2x)

Derivative of sqrt(1+cos2x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  ______________
\/ 1 + cos(2*x)
$$\sqrt{\cos{\left(2 x \right)} + 1}$$ 
sqrt(1 + cos(2*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. Let .

      3. The derivative of cosine is negative sine:

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

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      The result is:

    The result of the chain rule is:

  4. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   -sin(2*x)    
----------------
  ______________
\/ 1 + cos(2*x)
$$- \frac{\sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)} + 1}}$$
Simplify
The second derivative [src] 
 /                 2       \ 
 |              sin (2*x)  | 
-|2*cos(2*x) + ------------| 
 \             1 + cos(2*x)/ 
-----------------------------
         ______________      
       \/ 1 + cos(2*x)
$$- \frac{2 \cos{\left(2 x \right)} + \frac{\sin^{2}{\left(2 x \right)}}{\cos{\left(2 x \right)} + 1}}{\sqrt{\cos{\left(2 x \right)} + 1}}$$
Simplify
The third derivative [src] 
/                          2       \         
|     6*cos(2*x)      3*sin (2*x)  |         
|4 - ------------ - ---------------|*sin(2*x)
|    1 + cos(2*x)                 2|         
\                   (1 + cos(2*x)) /         
---------------------------------------------
                 ______________              
               \/ 1 + cos(2*x)
$$\frac{\left(4 - \frac{6 \cos{\left(2 x \right)}}{\cos{\left(2 x \right)} + 1} - \frac{3 \sin^{2}{\left(2 x \right)}}{\left(\cos{\left(2 x \right)} + 1\right)^{2}}\right) \sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)} + 1}}$$
Simplify
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