Mister Exam

Other calculators

sinx/(3-2cos(x))
The derivative of the function/ sinx/(3-2cos(x))

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

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   sin(x)   
------------
3 - 2*cos(x)
sin⁡(x)−2cos⁡(x)+3\frac{\sin{\left(x \right)}}{- 2 \cos{\left(x \right)} + 3} 
d /   sin(x)   \
--|------------|
dx\3 - 2*cos(x)/
ddxsin⁡(x)−2cos⁡(x)+3\frac{d}{d x} \frac{\sin{\left(x \right)}}{- 2 \cos{\left(x \right)} + 3}
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)=3−2cos⁡(x)g{\left(x \right)} = 3 - 2 \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 3−2cos⁡(x)3 - 2 \cos{\left(x \right)} term by term:

      1. The derivative of the constant 33 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: 2sin⁡(x)2 \sin{\left(x \right)}

      The result is: 2sin⁡(x)2 \sin{\left(x \right)}

    Now plug in to the quotient rule:

    (3−2cos⁡(x))cos⁡(x)−2sin⁡2(x)(3−2cos⁡(x))2\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}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-10102-2
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))
cos⁡(x)−2cos⁡(x)+3−2sin⁡2(x)(−2cos⁡(x)+3)2\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)
(−2(cos⁡(x)+4sin⁡2(x)2cos⁡(x)−3)2cos⁡(x)−3+1−4cos⁡(x)2cos⁡(x)−3)sin⁡(x)2cos⁡(x)−3\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)
−2(−1+12cos⁡(x)2cos⁡(x)−3+24sin⁡2(x)(2cos⁡(x)−3)2)sin⁡2(x)2cos⁡(x)−3−6(cos⁡(x)+4sin⁡2(x)2cos⁡(x)−3)cos⁡(x)2cos⁡(x)−3+cos⁡(x)+6sin⁡2(x)2cos⁡(x)−32cos⁡(x)−3\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
Derivative of sinx/(3-2cos(x))
    © Mister Exam – Calculator