Derivative of sinx/x^2

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sin(x)
------
   2  
  x

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

d /sin(x)\
--|------|
dx|   2  |
  \  x   /

$$\frac{d}{d x} \frac{\sin{\left(x \right)}}{x^{2}}$$

Transform

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 $g{\left(x \right)} = x^{2}$ .

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: $x^{2}$ goes to $2 x$
Now plug in to the quotient rule:

$\frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)}}{x^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(x)   2*sin(x)
------ - --------
   2         3   
  x         x

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

Simplify

The second derivative [src]

          4*cos(x)   6*sin(x)
-sin(x) - -------- + --------
             x           2   
                        x    
-----------------------------
               2             
              x

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

Simplify

The third derivative [src]

          24*sin(x)   6*sin(x)   18*cos(x)
-cos(x) - --------- + -------- + ---------
               3         x            2   
              x                      x    
------------------------------------------
                     2                    
                    x

$$\frac{- \cos{\left(x \right)} + \frac{6 \sin{\left(x \right)}}{x} + \frac{18 \cos{\left(x \right)}}{x^{2}} - \frac{24 \sin{\left(x \right)}}{x^{3}}}{x^{2}}$$

Simplify

The graph

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Identical expressions

Derivative of sinx/x^2

The graph:

Piecewise:

The solution