Derivative of cosx/x

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cos(x)
------
  x

$$\frac{\cos{\left(x \right)}}{x}$$

d /cos(x)\
--|------|
dx\  x   /

$$\frac{d}{d x} \frac{\cos{\left(x \right)}}{x}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of cosx/x

The graph:

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The solution