Derivative of sin(x)/((5*x))

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sin(x)
------
 5*x

$$\frac{\sin{\left(x \right)}}{5 x}$$

sin(x)/((5*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 $g{\left(x \right)} = 5 x$ .

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. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $5$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of sin(x)/((5*x))

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The solution