Derivative of x/sin(x)

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  x   
------
sin(x)

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /         2   \           
  |    2*cos (x)|   2*cos(x)
x*|1 + ---------| - --------
  |        2    |    sin(x) 
  \     sin (x) /           
----------------------------
           sin(x)

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of x/sin(x)

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The solution