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Similar expressions
sinx/(x)^(1/2)

The derivative of the function/ sin(x)/(x)^(1/2)

Derivative of sin(x)/(x)^(1/2)

The solution

You have entered [src]

sin(x)
------
  ___ 
\/ x

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

sin(x)/sqrt(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(x)   sin(x)
------ - ------
  ___       3/2
\/ x     2*x

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

Simplify

The second derivative [src]

          cos(x)   3*sin(x)
-sin(x) - ------ + --------
            x           2  
                     4*x   
---------------------------
             ___           
           \/ x

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

Simplify

The third derivative [src]

          15*sin(x)   3*sin(x)   9*cos(x)
-cos(x) - --------- + -------- + --------
                3       2*x           2  
             8*x                   4*x   
-----------------------------------------
                    ___                  
                  \/ x

$$\frac{- \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{2 x} + \frac{9 \cos{\left(x \right)}}{4 x^{2}} - \frac{15 \sin{\left(x \right)}}{8 x^{3}}}{\sqrt{x}}$$

Simplify

The graph

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Identical expressions

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Derivative of sin(x)/(x)^(1/2)

The graph:

Piecewise:

The solution