Derivative of sqrt(cos(x))

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  ________
\/ cos(x)

$$\sqrt{\cos{\left(x \right)}}$$

sqrt(cos(x))

Detail solution

Let $u = \cos{\left(x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  -sin(x)   
------------
    ________
2*\/ cos(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

 /         2   \        
 |    3*sin (x)|        
-|2 + ---------|*sin(x) 
 |        2    |        
 \     cos (x) /        
------------------------
          ________      
      8*\/ cos(x)

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

Simplify

The graph

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Derivative of sqrt(cos(x))

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The solution