Derivative of y=(x-cosx)/sinx

The solution

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x - cos(x)
----------
  sin(x)

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

(x - cos(x))/sin(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x - \cos{\left(x \right)}$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $\sin{\left(x \right)}$
  The result is: $\sin{\left(x \right)} + 1$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                   /         2   \                /         2   \                    
                   |    2*cos (x)|                |    6*cos (x)|                    
                 3*|1 + ---------|*(1 + sin(x))   |5 + ---------|*(x - cos(x))*cos(x)
          2        |        2    |                |        2    |                    
     3*cos (x)     \     sin (x) /                \     sin (x) /                    
-1 - --------- + ------------------------------ - -----------------------------------
         2                   sin(x)                                2                 
      sin (x)                                                   sin (x)

$$\frac{3 \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \left(\sin{\left(x \right)} + 1\right)}{\sin{\left(x \right)}} - \frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \left(x - \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 1 - \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$

Simplify

The graph

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Derivative of y=(x-cosx)/sinx

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