Mister Exam

Derivative of y=(sinx)/(1-cosx)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  sin(x)  
----------
1 - cos(x)
$$\frac{\sin{\left(x \right)}}{- \cos{\left(x \right)} + 1}$$
d /  sin(x)  \
--|----------|
dx\1 - cos(x)/
$$\frac{d}{d x} \frac{\sin{\left(x \right)}}{- \cos{\left(x \right)} + 1}$$
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of sine is cosine:

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant 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:

        So, the result is:

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                   2      
  cos(x)        sin (x)   
---------- - -------------
1 - cos(x)               2
             (1 - cos(x)) 
$$\frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1} - \frac{\sin^{2}{\left(x \right)}}{\left(- \cos{\left(x \right)} + 1\right)^{2}}$$
The second derivative [src]
/          2                           \       
|     2*sin (x)                        |       
|    ----------- + cos(x)              |       
|    -1 + cos(x)              2*cos(x) |       
|1 - -------------------- - -----------|*sin(x)
\        -1 + cos(x)        -1 + cos(x)/       
-----------------------------------------------
                  -1 + cos(x)                  
$$\frac{\left(1 - \frac{\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{\cos{\left(x \right)} - 1} - \frac{2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1}$$
The third derivative [src]
                      /                          2      \                                           
                 2    |       6*cos(x)      6*sin (x)   |     /      2             \                
              sin (x)*|-1 + ----------- + --------------|     | 2*sin (x)          |                
      2               |     -1 + cos(x)                2|   3*|----------- + cos(x)|*cos(x)         
 3*sin (x)            \                   (-1 + cos(x)) /     \-1 + cos(x)         /                
----------- - ------------------------------------------- - ------------------------------- + cos(x)
-1 + cos(x)                   -1 + cos(x)                             -1 + cos(x)                   
----------------------------------------------------------------------------------------------------
                                            -1 + cos(x)                                             
$$\frac{- \frac{\left(-1 + \frac{6 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{6 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}}\right) \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)} - \frac{3 \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{\cos{\left(x \right)} - 1}$$
The graph
Derivative of y=(sinx)/(1-cosx)