Derivative of sqrt(1-cos2x)

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  ______________
\/ 1 - cos(2*x)

$$\sqrt{1 - \cos{\left(2 x \right)}}$$

sqrt(1 - cos(2*x))

Detail solution

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

$\frac{\sin{\left(2 x \right)}}{\sqrt{1 - \cos{\left(2 x \right)}}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

    sin(2*x)    
----------------
  ______________
\/ 1 - cos(2*x)

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

Simplify

The second derivative [src]

                 2       
              sin (2*x)  
2*cos(2*x) - ------------
             1 - cos(2*x)
-------------------------
       ______________    
     \/ 1 - cos(2*x)

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

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)

$$\frac{\left(-4 - \frac{6 \cos{\left(2 x \right)}}{1 - \cos{\left(2 x \right)}} + \frac{3 \sin^{2}{\left(2 x \right)}}{\left(1 - \cos{\left(2 x \right)}\right)^{2}}\right) \sin{\left(2 x \right)}}{\sqrt{1 - \cos{\left(2 x \right)}}}$$

Simplify

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Derivative of sqrt(1-cos2x)

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The solution