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Derivative of sinx/(3-2cos(x))
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Identical expressions
sinx/(three -2cos(x))
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sinus of x divide by (three minus 2 co sinus of e of (x))
sinx/3-2cosx
sinx divide by (3-2cos(x))
Similar expressions
sinx/(3+2cos(x))
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The derivative of the function/ sinx/(3-2cos(x))

Derivative of sinx/(3-2cos(x))

The solution

You have entered [src]

   sin(x)   
------------
3 - 2*cos(x)

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

d /   sin(x)   \
--|------------|
dx\3 - 2*cos(x)/

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

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 $g{\left(x \right)} = 3 - 2 \cos{\left(x \right)}$ .

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 $3 - 2 \cos{\left(x \right)}$ term by term:
  1. The derivative of the constant $3$ 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: $2 \sin{\left(x \right)}$
  The result is: $2 \sin{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{\left(3 - 2 \cos{\left(x \right)}\right) \cos{\left(x \right)} - 2 \sin^{2}{\left(x \right)}}{\left(3 - 2 \cos{\left(x \right)}\right)^{2}}$
Now simplify:

$\frac{3 \cos{\left(x \right)} - 2}{\left(2 \cos{\left(x \right)} - 3\right)^{2}}$

The answer is:

$\frac{3 \cos{\left(x \right)} - 2}{\left(2 \cos{\left(x \right)} - 3\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                       2      
   cos(x)         2*sin (x)   
------------ - ---------------
3 - 2*cos(x)                 2
               (3 - 2*cos(x))

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

Simplify

The second derivative [src]

/                      /       2              \\       
|                      |  4*sin (x)           ||       
|                    2*|------------- + cos(x)||       
|       4*cos(x)       \-3 + 2*cos(x)         /|       
|1 - ------------- - --------------------------|*sin(x)
\    -3 + 2*cos(x)         -3 + 2*cos(x)       /       
-------------------------------------------------------
                     -3 + 2*cos(x)

$$\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 3}\right)}{2 \cos{\left(x \right)} - 3} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 3}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 3}$$

Simplify

The third derivative [src]

                                                              /                              2      \         
                  /       2              \               2    |       12*cos(x)        24*sin (x)   |         
                  |  4*sin (x)           |          2*sin (x)*|-1 + ------------- + ----------------|         
       2        6*|------------- + cos(x)|*cos(x)             |     -3 + 2*cos(x)                  2|         
  6*sin (x)       \-3 + 2*cos(x)         /                    \                     (-3 + 2*cos(x)) /         
------------- - --------------------------------- - ------------------------------------------------- + cos(x)
-3 + 2*cos(x)             -3 + 2*cos(x)                               -3 + 2*cos(x)                           
--------------------------------------------------------------------------------------------------------------
                                                -3 + 2*cos(x)

$$\frac{- \frac{2 \left(-1 + \frac{12 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 3} + \frac{24 \sin^{2}{\left(x \right)}}{\left(2 \cos{\left(x \right)} - 3\right)^{2}}\right) \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 3} - \frac{6 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 3}\right) \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 3} + \cos{\left(x \right)} + \frac{6 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 3}}{2 \cos{\left(x \right)} - 3}$$

Simplify

The graph

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Derivative of sinx/(3-2cos(x))

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Piecewise:

The solution