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Limit of the function:
sqrt(sin(x))/x
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sqrt(sin(x))/x
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sqrt(sin(x))/x
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sqrt(sinx)/x

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

Derivative of sqrt(sin(x))/x

The solution

You have entered [src]

  ________
\/ sin(x) 
----------
    x

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

sqrt(sin(x))/x

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $\frac{\cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    ________                 
  \/ sin(x)        cos(x)    
- ---------- + --------------
       2             ________
      x        2*x*\/ sin(x)

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

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

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

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

$$\frac{\frac{\left(2 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{8 \sqrt{\sin{\left(x \right)}}} + \frac{3 \left(2 \sqrt{\sin{\left(x \right)}} + \frac{\cos^{2}{\left(x \right)}}{\sin^{\frac{3}{2}}{\left(x \right)}}\right)}{4 x} + \frac{3 \cos{\left(x \right)}}{x^{2} \sqrt{\sin{\left(x \right)}}} - \frac{6 \sqrt{\sin{\left(x \right)}}}{x^{3}}}{x}$$

Simplify

The graph

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Derivative of sqrt(sin(x))/x

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The solution