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Identical expressions
cos(x)/((five *x^ two))
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co sinus of e of (x) divide by ((five multiply by x to the power of two))
cos(x)/((5*x2))
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cos(x)/((5*x²))
cos(x)/((5*x to the power of 2))
cos(x)/((5x^2))
cos(x)/((5x2))
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Similar expressions
cosx/((5*x^2))

The derivative of the function/ cos(x)/((5*x^2))

Derivative of cos(x)/((5*x^2))

The solution

You have entered [src]

cos(x)
------
    2 
 5*x

$$\frac{\cos{\left(x \right)}}{5 x^{2}}$$

cos(x)/((5*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 $g{\left(x \right)} = 5 x^{2}$ .

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. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  So, the result is: $10 x$
Now plug in to the quotient rule:

$\frac{- 5 x^{2} \sin{\left(x \right)} - 10 x \cos{\left(x \right)}}{25 x^{4}}$
Now simplify:

$- \frac{x \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{5 x^{3}}$

The answer is:

$- \frac{x \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{5 x^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   1            2*cos(x)
- ----*sin(x) - --------
     2               3  
  5*x             5*x

$$- \frac{1}{5 x^{2}} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{5 x^{3}}$$

Simplify

The second derivative [src]

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

$$\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \cos{\left(x \right)}}{x^{2}}}{5 x^{2}}$$

Simplify

The third derivative [src]

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

$$\frac{\sin{\left(x \right)} + \frac{6 \cos{\left(x \right)}}{x} - \frac{18 \sin{\left(x \right)}}{x^{2}} - \frac{24 \cos{\left(x \right)}}{x^{3}}}{5 x^{2}}$$

Simplify

The graph

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Derivative of cos(x)/((5*x^2))

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The solution