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sinx/(2+3cos^2x)

Derivative of sinx/(2+3cos^2x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
    sin(x)   
-------------
         2   
2 + 3*cos (x)
$$\frac{\sin{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2}$$
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. Let .

        2. Apply the power rule: goes to

        3. Then, apply the chain rule. Multiply by :

          1. The derivative of cosine is negative sine:

          The result of the chain rule is:

        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)      6*sin (x)*cos(x)
------------- + ----------------
         2                     2
2 + 3*cos (x)   /         2   \ 
                \2 + 3*cos (x)/ 
$$\frac{\cos{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2} + \frac{6 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(3 \cos^{2}{\left(x \right)} + 2\right)^{2}}$$
The second derivative [src]
/       /                          2       2   \                \       
|       |   2         2      12*cos (x)*sin (x)|                |       
|     6*|cos (x) - sin (x) + ------------------|                |       
|       |                               2      |           2    |       
|       \                      2 + 3*cos (x)   /     12*cos (x) |       
|-1 + ------------------------------------------ + -------------|*sin(x)
|                            2                              2   |       
\                   2 + 3*cos (x)                  2 + 3*cos (x)/       
------------------------------------------------------------------------
                                      2                                 
                             2 + 3*cos (x)                              
$$\frac{\left(-1 + \frac{6 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{12 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2}\right)}{3 \cos^{2}{\left(x \right)} + 2} + \frac{12 \cos^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2}\right) \sin{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2}$$
The third derivative [src]
/                                                                              /           2               2              2       2   \\       
|                        /                          2       2   \         2    |      9*cos (x)       9*sin (x)     54*cos (x)*sin (x)||       
|                        |   2         2      12*cos (x)*sin (x)|   24*sin (x)*|1 - ------------- + ------------- - ------------------||       
|                     18*|cos (x) - sin (x) + ------------------|              |             2               2                      2 ||       
|             2          |                               2      |              |    2 + 3*cos (x)   2 + 3*cos (x)    /         2   \  ||       
|       18*sin (x)       \                      2 + 3*cos (x)   /              \                                     \2 + 3*cos (x)/  /|       
|-1 - ------------- + ------------------------------------------- - -------------------------------------------------------------------|*cos(x)
|              2                              2                                                         2                              |       
\     2 + 3*cos (x)                  2 + 3*cos (x)                                             2 + 3*cos (x)                           /       
-----------------------------------------------------------------------------------------------------------------------------------------------
                                                                          2                                                                    
                                                                 2 + 3*cos (x)                                                                 
$$\frac{\left(-1 + \frac{18 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{12 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2}\right)}{3 \cos^{2}{\left(x \right)} + 2} - \frac{24 \left(1 + \frac{9 \sin^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2} - \frac{9 \cos^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2} - \frac{54 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(3 \cos^{2}{\left(x \right)} + 2\right)^{2}}\right) \sin^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2} - \frac{18 \sin^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2}\right) \cos{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2}$$
The graph
Derivative of sinx/(2+3cos^2x)