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sqrt(sin(x))/x
The derivative of the function/ sqrt(sin(x))/x

Derivative of sqrt(sin(x))/x

Function f() - derivative -N order at the point
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The solution

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

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

    f(x)=sin⁡(x)f{\left(x \right)} = \sqrt{\sin{\left(x \right)}} and g(x)=xg{\left(x \right)} = x.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

    2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

    3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

      1. The derivative of sine is cosine:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      The result of the chain rule is:

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

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    Now plug in to the quotient rule:

    xcos⁡(x)2sin⁡(x)−sin⁡(x)x2\frac{\frac{x \cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}} - \sqrt{\sin{\left(x \right)}}}{x^{2}}

  2. Now simplify:

    xcos⁡(x)2−sin⁡(x)x2sin⁡(x)\frac{\frac{x \cos{\left(x \right)}}{2} - \sin{\left(x \right)}}{x^{2} \sqrt{\sin{\left(x \right)}}}

The answer is:

xcos⁡(x)2−sin⁡(x)x2sin⁡(x)\frac{\frac{x \cos{\left(x \right)}}{2} - \sin{\left(x \right)}}{x^{2} \sqrt{\sin{\left(x \right)}}}

The graph
02468-8-6-4-2-1010-2020
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    ________                 
  \/ sin(x)        cos(x)    
- ---------- + --------------
       2             ________
      x        2*x*\/ sin(x)
cos⁡(x)2xsin⁡(x)−sin⁡(x)x2\frac{\cos{\left(x \right)}}{2 x \sqrt{\sin{\left(x \right)}}} - \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}
Simplify
The second derivative [src] 
    ________       ________        2                    
  \/ sin(x)    2*\/ sin(x)      cos (x)        cos(x)   
- ---------- + ------------ - ----------- - ------------
      2              2             3/2          ________
                    x         4*sin   (x)   x*\/ sin(x) 
--------------------------------------------------------
                           x
−sin⁡(x)2−cos⁡2(x)4sin⁡32(x)−cos⁡(x)xsin⁡(x)+2sin⁡(x)x2x\frac{- \frac{\sqrt{\sin{\left(x \right)}}}{2} - \frac{\cos^{2}{\left(x \right)}}{4 \sin^{\frac{3}{2}}{\left(x \right)}} - \frac{\cos{\left(x \right)}}{x \sqrt{\sin{\left(x \right)}}} + \frac{2 \sqrt{\sin{\left(x \right)}}}{x^{2}}}{x}
Simplify
The third derivative [src] 
                   /                   2    \                   /         2   \       
                   |    ________    cos (x) |                   |    3*cos (x)|       
                 3*|2*\/ sin(x)  + ---------|                   |2 + ---------|*cos(x)
      ________     |                  3/2   |                   |        2    |       
  6*\/ sin(x)      \               sin   (x)/      3*cos(x)     \     sin (x) /       
- ------------ + ---------------------------- + ------------- + ----------------------
        3                    4*x                 2   ________            ________     
       x                                        x *\/ sin(x)         8*\/ sin(x)      
--------------------------------------------------------------------------------------
                                          x
(2+3cos⁡2(x)sin⁡2(x))cos⁡(x)8sin⁡(x)+3(2sin⁡(x)+cos⁡2(x)sin⁡32(x))4x+3cos⁡(x)x2sin⁡(x)−6sin⁡(x)x3x\frac{\frac{\left(2 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{8 \sqrt{\sin{\left(x \right)}}} + \frac{3 \left(2 \sqrt{\sin{\left(x \right)}} + \frac{\cos^{2}{\left(x \right)}}{\sin^{\frac{3}{2}}{\left(x \right)}}\right)}{4 x} + \frac{3 \cos{\left(x \right)}}{x^{2} \sqrt{\sin{\left(x \right)}}} - \frac{6 \sqrt{\sin{\left(x \right)}}}{x^{3}}}{x}
Simplify
The graph
Derivative of sqrt(sin(x))/x
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