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cosx/x^2

The derivative of the function/ cos(x)/x^2

Derivative of cos(x)/x^2

The solution

You have entered [src]

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

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

cos(x)/x^2

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of cos(x)/x^2

The graph:

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The solution