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

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)/

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
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:

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

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

The answer is:

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

The graph

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))

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

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)

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

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)

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution