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Identical expressions
(sqrt(one +cos^2x))^(one / four)
( square root of (1 plus co sinus of e of squared x)) to the power of (1 divide by 4)
( square root of (one plus co sinus of e of squared x)) to the power of (one divide by four)
(√(1+cos^2x))^(1/4)
(sqrt(1+cos2x))(1/4)
sqrt1+cos2x1/4
(sqrt(1+cos²x))^(1/4)
(sqrt(1+cos to the power of 2x)) to the power of (1/4)
sqrt1+cos^2x^1/4
(sqrt(1+cos^2x))^(1 divide by 4)
Similar expressions
(sqrt(1-cos^2x))^(1/4)

The derivative of the function/ (sqrt(1+cos^2x))^(1/4)

Derivative of (sqrt(1+cos^2x))^(1/4)

The solution

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    __________________
   /    _____________ 
4 /    /        2     
\/   \/  1 + cos (x)

$$\sqrt[4]{\sqrt{\cos^{2}{\left(x \right)} + 1}}$$

(sqrt(1 + cos(x)^2))^(1/4)

Detail solution

Let $u = \sqrt{\cos^{2}{\left(x \right)} + 1}$ .
Apply the power rule: $\sqrt[4]{u}$ goes to $\frac{1}{4 u^{\frac{3}{4}}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{\cos^{2}{\left(x \right)} + 1}$ :
1. Let $u = \cos^{2}{\left(x \right)} + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\cos^{2}{\left(x \right)} + 1\right)$ :
  1. Differentiate $\cos^{2}{\left(x \right)} + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. Let $u = \cos{\left(x \right)}$ .
    3. Apply the power rule: $u^{2}$ goes to $2 u$
    4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      The result of the chain rule is:
      
      $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
    The result is: $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1}}$
The result of the chain rule is:

$- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{4 \left(\cos^{2}{\left(x \right)} + 1\right)^{\frac{7}{8}}}$
Now simplify:

$- \frac{2^{\frac{7}{8}} \sin{\left(2 x \right)}}{8 \left(\cos{\left(2 x \right)} + 3\right)^{\frac{7}{8}}}$

The answer is:

$- \frac{2^{\frac{7}{8}} \sin{\left(2 x \right)}}{8 \left(\cos{\left(2 x \right)} + 3\right)^{\frac{7}{8}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    _____________               
 8 /        2                   
-\/  1 + cos (x) *cos(x)*sin(x) 
--------------------------------
          /       2   \         
        4*\1 + cos (x)/

$$- \frac{\sqrt[8]{\cos^{2}{\left(x \right)} + 1} \sin{\left(x \right)} \cos{\left(x \right)}}{4 \left(\cos^{2}{\left(x \right)} + 1\right)}$$

Simplify

The second derivative [src]

                               2       2   
       2           2      7*cos (x)*sin (x)
- 4*cos (x) + 4*sin (x) - -----------------
                                    2      
                             1 + cos (x)   
-------------------------------------------
                            7/8            
               /       2   \               
            16*\1 + cos (x)/

$$\frac{4 \sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)} - \frac{7 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 1}}{16 \left(\cos^{2}{\left(x \right)} + 1\right)^{\frac{7}{8}}}$$

Simplify

The third derivative [src]

/             2                  2                2       2   \              
|       21*cos (x)         21*sin (x)      105*cos (x)*sin (x)|              
|1 - ---------------- + ---------------- - -------------------|*cos(x)*sin(x)
|       /       2   \      /       2   \                    2 |              
|    16*\1 + cos (x)/   16*\1 + cos (x)/       /       2   \  |              
\                                           64*\1 + cos (x)/  /              
-----------------------------------------------------------------------------
                                            7/8                              
                               /       2   \                                 
                               \1 + cos (x)/

$$\frac{\left(1 + \frac{21 \sin^{2}{\left(x \right)}}{16 \left(\cos^{2}{\left(x \right)} + 1\right)} - \frac{21 \cos^{2}{\left(x \right)}}{16 \left(\cos^{2}{\left(x \right)} + 1\right)} - \frac{105 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{64 \left(\cos^{2}{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{\frac{7}{8}}}$$

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Derivative of (sqrt(1+cos^2x))^(1/4)

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The solution