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The derivative of the function/ sqrt(sinx/2)

Derivative of sqrt(sinx/2)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
    ________
   / sin(x) 
  /  ------ 
\/     2
$$\sqrt{\frac{\sin{\left(x \right)}}{2}}$$ 
sqrt(sin(x)/2)
Detail solution

  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. The derivative of sine is cosine:

      So, the result is:

    The result of the chain rule is:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  ___   ________       
\/ 2 *\/ sin(x)        
----------------*cos(x)
       2               
-----------------------
        2*sin(x)
$$\frac{\frac{\sqrt{2} \sqrt{\sin{\left(x \right)}}}{2} \cos{\left(x \right)}}{2 \sin{\left(x \right)}}$$
Simplify
The second derivative [src] 
       /                   2    \ 
   ___ |    ________    cos (x) | 
-\/ 2 *|2*\/ sin(x)  + ---------| 
       |                  3/2   | 
       \               sin   (x)/ 
----------------------------------
                8
$$- \frac{\sqrt{2} \left(2 \sqrt{\sin{\left(x \right)}} + \frac{\cos^{2}{\left(x \right)}}{\sin^{\frac{3}{2}}{\left(x \right)}}\right)}{8}$$
Simplify
The third derivative [src] 
      /         2   \       
  ___ |    3*cos (x)|       
\/ 2 *|2 + ---------|*cos(x)
      |        2    |       
      \     sin (x) /       
----------------------------
            ________        
       16*\/ sin(x)
$$\frac{\sqrt{2} \left(2 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{16 \sqrt{\sin{\left(x \right)}}}$$
Simplify
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