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Similar expressions
4sqrt(1-cos^2x)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
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:
  
  $- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1}}$
So, the result is: $- \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /          2             2            2       2   \              
  |     3*cos (x)     3*sin (x)    3*cos (x)*sin (x)|              
4*|4 - ----------- + ----------- - -----------------|*cos(x)*sin(x)
  |           2             2                     2 |              
  |    1 + cos (x)   1 + cos (x)     /       2   \  |              
  \                                  \1 + cos (x)/  /              
-------------------------------------------------------------------
                             _____________                         
                            /        2                             
                          \/  1 + cos (x)

$$\frac{4 \left(4 + \frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 1} - \frac{3 \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 1} - \frac{3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1}}$$

Simplify

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

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The solution