Mister Exam

Other calculators

y=(sinx)/(1-cosx)
The derivative of the function/ y=(sinx)/(1-cosx)

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)
sin⁡(x)−cos⁡(x)+1\frac{\sin{\left(x \right)}}{- \cos{\left(x \right)} + 1} 
d /  sin(x)  \
--|----------|
dx\1 - cos(x)/
ddxsin⁡(x)−cos⁡(x)+1\frac{d}{d x} \frac{\sin{\left(x \right)}}{- \cos{\left(x \right)} + 1}
Transform
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=1−cos⁡(x)g{\left(x \right)} = 1 - \cos{\left(x \right)}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate 1−cos⁡(x)1 - \cos{\left(x \right)} term by term:

      1. The derivative of the constant 11 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:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        So, the result is: sin⁡(x)\sin{\left(x \right)}

      The result is: sin⁡(x)\sin{\left(x \right)}

    Now plug in to the quotient rule:

    (1−cos⁡(x))cos⁡(x)−sin⁡2(x)(1−cos⁡(x))2\frac{\left(1 - \cos{\left(x \right)}\right) \cos{\left(x \right)} - \sin^{2}{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}}

  2. Now simplify:

    1cos⁡(x)−1\frac{1}{\cos{\left(x \right)} - 1}

The answer is:

1cos⁡(x)−1\frac{1}{\cos{\left(x \right)} - 1}

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                   2      
  cos(x)        sin (x)   
---------- - -------------
1 - cos(x)               2
             (1 - cos(x))
cos⁡(x)−cos⁡(x)+1−sin⁡2(x)(−cos⁡(x)+1)2\frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1} - \frac{\sin^{2}{\left(x \right)}}{\left(- \cos{\left(x \right)} + 1\right)^{2}}
Simplify
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)
(1−cos⁡(x)+2sin⁡2(x)cos⁡(x)−1cos⁡(x)−1−2cos⁡(x)cos⁡(x)−1)sin⁡(x)cos⁡(x)−1\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}
Simplify
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)
−(−1+6cos⁡(x)cos⁡(x)−1+6sin⁡2(x)(cos⁡(x)−1)2)sin⁡2(x)cos⁡(x)−1+cos⁡(x)−3(cos⁡(x)+2sin⁡2(x)cos⁡(x)−1)cos⁡(x)cos⁡(x)−1+3sin⁡2(x)cos⁡(x)−1cos⁡(x)−1\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}
Simplify
The graph
Derivative of y=(sinx)/(1-cosx)
    © Mister Exam – Calculator