Derivative of cox^2x/sin2x

The solution

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

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

(cos(x)^2*x)/sin(2*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x \cos^{2}{\left(x \right)}$ and $g{\left(x \right)} = \sin{\left(2 x \right)}$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                                                                                                                                                     /         2     \         \
  |                                                                                                                                                                2    |    6*cos (2*x)|         |
  |                                                                                                                                                         4*x*cos (x)*|5 + -----------|*cos(2*x)|
  |                            /  /   2         2   \                  \              /         2     \                                                                 |        2      |         |
  |       2           2      6*\x*\sin (x) - cos (x)/ - 2*cos(x)*sin(x)/*cos(2*x)     |    2*cos (2*x)|                                                                 \     sin (2*x) /         |
2*|- 3*cos (x) + 3*sin (x) - ---------------------------------------------------- - 6*|1 + -----------|*(-cos(x) + 2*x*sin(x))*cos(x) + 4*x*cos(x)*sin(x) - --------------------------------------|
  |                                                sin(2*x)                           |        2      |                                                                    sin(2*x)               |
  \                                                                                   \     sin (2*x) /                                                                                           /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                              sin(2*x)

$$\frac{2 \left(- \frac{4 x \left(5 + \frac{6 \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right) \cos^{2}{\left(x \right)} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 4 x \sin{\left(x \right)} \cos{\left(x \right)} - 6 \left(1 + \frac{2 \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right) \left(2 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)} - \frac{6 \left(x \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 3 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right)}{\sin{\left(2 x \right)}}$$

Simplify

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Derivative of cox^2x/sin2x

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The solution