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Similar expressions
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The derivative of the function/ y=sqrt(x/2-sin(x/2))

Derivative of y=sqrt(x/2-sin(x/2))

The solution

You have entered [src]

    ____________
   / x      /x\ 
  /  - - sin|-| 
\/   2      \2/

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

sqrt(x/2 - sin(x/2))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          /x\   
       cos|-|   
   1      \2/   
   - - ------   
   4     4      
----------------
    ____________
   / x      /x\ 
  /  - - sin|-| 
\/   2      \2/

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

Simplify

The second derivative [src]

                        2
           /        /x\\ 
           |-1 + cos|-|| 
     /x\   \        \2// 
2*sin|-| - --------------
     \2/     x      /x\  
             - - sin|-|  
             2      \2/  
-------------------------
          ____________   
         / x      /x\    
   16*  /  - - sin|-|    
      \/   2      \2/

$$\frac{2 \sin{\left(\frac{x}{2} \right)} - \frac{\left(\cos{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\frac{x}{2} - \sin{\left(\frac{x}{2} \right)}}}{16 \sqrt{\frac{x}{2} - \sin{\left(\frac{x}{2} \right)}}}$$

Simplify

The third derivative [src]

                          3                         
             /        /x\\      /        /x\\    /x\
           3*|-1 + cos|-||    6*|-1 + cos|-||*sin|-|
     /x\     \        \2//      \        \2//    \2/
4*cos|-| - ---------------- + ----------------------
     \2/                2           x      /x\      
            /x      /x\\            - - sin|-|      
            |- - sin|-||            2      \2/      
            \2      \2//                            
----------------------------------------------------
                       ____________                 
                      / x      /x\                  
                64*  /  - - sin|-|                  
                   \/   2      \2/

$$\frac{4 \cos{\left(\frac{x}{2} \right)} + \frac{6 \left(\cos{\left(\frac{x}{2} \right)} - 1\right) \sin{\left(\frac{x}{2} \right)}}{\frac{x}{2} - \sin{\left(\frac{x}{2} \right)}} - \frac{3 \left(\cos{\left(\frac{x}{2} \right)} - 1\right)^{3}}{\left(\frac{x}{2} - \sin{\left(\frac{x}{2} \right)}\right)^{2}}}{64 \sqrt{\frac{x}{2} - \sin{\left(\frac{x}{2} \right)}}}$$

Simplify

The graph

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Derivative of y=sqrt(x/2-sin(x/2))

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