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(x*sinx)/(one -cosx)
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Similar expressions
(x*sinx)/(1+cosx)
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The derivative of the function/ (x*sinx)/(1-cosx)

Derivative of (x*sinx)/(1-cosx)

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x \sin{\left(x \right)}$ and $g{\left(x \right)} = 1 - \cos{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $1 - \cos{\left(x \right)}$ term by term:
  1. The derivative of the constant $1$ is zero.
  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: $\sin{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Similar expressions

Derivative of (x*sinx)/(1-cosx)

The graph:

Piecewise:

The solution