Derivative of sqrt(cos(2x))

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  __________
\/ cos(2*x)

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

sqrt(cos(2*x))

Detail solution

Let $u = \cos{\left(2 x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x \right)}$ :
1. Let $u = 2 x$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $2$
  The result of the chain rule is:
  
  $- 2 \sin{\left(2 x \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of sqrt(cos(2x))

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The solution