Derivative of sen3x/x

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sin(3*x)
--------
   x

$$\frac{\sin{\left(3 x \right)}}{x}$$

sin(3*x)/x

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 3 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
  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: $3$
  The result of the chain rule is:
  
  $3 \cos{\left(3 x \right)}$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

              6*cos(3*x)   2*sin(3*x)
-9*sin(3*x) - ---------- + ----------
                  x             2    
                               x     
-------------------------------------
                  x

$$\frac{- 9 \sin{\left(3 x \right)} - \frac{6 \cos{\left(3 x \right)}}{x} + \frac{2 \sin{\left(3 x \right)}}{x^{2}}}{x}$$

Simplify

The third derivative [src]

  /              2*sin(3*x)   6*cos(3*x)   9*sin(3*x)\
3*|-9*cos(3*x) - ---------- + ---------- + ----------|
  |                   3            2           x     |
  \                  x            x                  /
------------------------------------------------------
                          x

$$\frac{3 \left(- 9 \cos{\left(3 x \right)} + \frac{9 \sin{\left(3 x \right)}}{x} + \frac{6 \cos{\left(3 x \right)}}{x^{2}} - \frac{2 \sin{\left(3 x \right)}}{x^{3}}\right)}{x}$$

Simplify

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Derivative of sen3x/x

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The solution