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Similar expressions
sinx/(2-3cos^2x)

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

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

The solution

You have entered [src]

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

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

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

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 $3 \cos^{2}{\left(x \right)} + 2$ term by term:
  1. The derivative of the constant $2$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \cos{\left(x \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      The result of the chain rule is:
      
      $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
    So, the result is: $- 6 \sin{\left(x \right)} \cos{\left(x \right)}$
  The result is: $- 6 \sin{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{\left(3 \cos^{2}{\left(x \right)} + 2\right) \cos{\left(x \right)} + 6 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(3 \cos^{2}{\left(x \right)} + 2\right)^{2}}$
Now simplify:

$\frac{\left(3 \sin^{2}{\left(x \right)} + 5\right) \cos{\left(x \right)}}{\left(3 \cos^{2}{\left(x \right)} + 2\right)^{2}}$

The answer is:

$\frac{\left(3 \sin^{2}{\left(x \right)} + 5\right) \cos{\left(x \right)}}{\left(3 \cos^{2}{\left(x \right)} + 2\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                     2          
    cos(x)      6*sin (x)*cos(x)
------------- + ----------------
         2                     2
2 + 3*cos (x)   /         2   \ 
                \2 + 3*cos (x)/

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

Simplify

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)

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

Simplify

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)

$$\frac{\left(-1 + \frac{18 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{12 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2}\right)}{3 \cos^{2}{\left(x \right)} + 2} - \frac{24 \left(1 + \frac{9 \sin^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2} - \frac{9 \cos^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2} - \frac{54 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(3 \cos^{2}{\left(x \right)} + 2\right)^{2}}\right) \sin^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2} - \frac{18 \sin^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2}\right) \cos{\left(x \right)}}{3 \cos^{2}{\left(x \right)} + 2}$$

Simplify

The graph

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

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Piecewise:

The solution