Derivative of 2cost/sin2t-sint

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2*cos(t)         
-------- - sin(t)
sin(2*t)

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

(2*cos(t))/sin(2*t) - sin(t)

Detail solution

Differentiate $- \sin{\left(t \right)} + \frac{2 \cos{\left(t \right)}}{\sin{\left(2 t \right)}}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d t} \frac{f{\left(t \right)}}{g{\left(t \right)}} = \frac{- f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}}{g^{2}{\left(t \right)}}$
  
  $f{\left(t \right)} = 2 \cos{\left(t \right)}$ and $g{\left(t \right)} = \sin{\left(2 t \right)}$ .
  
  To find $\frac{d}{d t} f{\left(t \right)}$ :
  1. 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:
      
      $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
    So, the result is: $- 2 \sin{\left(t \right)}$
  To find $\frac{d}{d t} g{\left(t \right)}$ :
  1. Let $u = 2 t$ .
  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 $\frac{d}{d t} 2 t$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $t$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $2 \cos{\left(2 t \right)}$
  Now plug in to the quotient rule:
  
  $\frac{- 2 \sin{\left(t \right)} \sin{\left(2 t \right)} - 4 \cos{\left(t \right)} \cos{\left(2 t \right)}}{\sin^{2}{\left(2 t \right)}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of sine is cosine:
    
    $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
  So, the result is: $- \cos{\left(t \right)}$
The result is: $\frac{- 2 \sin{\left(t \right)} \sin{\left(2 t \right)} - 4 \cos{\left(t \right)} \cos{\left(2 t \right)}}{\sin^{2}{\left(2 t \right)}} - \cos{\left(t \right)}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

          2*sin(t)   4*cos(t)*cos(2*t)
-cos(t) - -------- - -----------------
          sin(2*t)          2         
                         sin (2*t)

$$- \frac{2 \sin{\left(t \right)}}{\sin{\left(2 t \right)}} - \cos{\left(t \right)} - \frac{4 \cos{\left(t \right)} \cos{\left(2 t \right)}}{\sin^{2}{\left(2 t \right)}}$$

Simplify

The second derivative [src]

                                     2                     
6*cos(t)   8*cos(2*t)*sin(t)   16*cos (2*t)*cos(t)         
-------- + ----------------- + ------------------- + sin(t)
sin(2*t)          2                    3                   
               sin (2*t)            sin (2*t)

$$\sin{\left(t \right)} + \frac{8 \sin{\left(t \right)} \cos{\left(2 t \right)}}{\sin^{2}{\left(2 t \right)}} + \frac{6 \cos{\left(t \right)}}{\sin{\left(2 t \right)}} + \frac{16 \cos{\left(t \right)} \cos^{2}{\left(2 t \right)}}{\sin^{3}{\left(2 t \right)}}$$

Simplify

The third derivative [src]

                    3                                          2                     
  22*sin(t)   96*cos (2*t)*cos(t)   68*cos(t)*cos(2*t)   48*cos (2*t)*sin(t)         
- --------- - ------------------- - ------------------ - ------------------- + cos(t)
   sin(2*t)           4                    2                     3                   
                   sin (2*t)            sin (2*t)             sin (2*t)

$$- \frac{22 \sin{\left(t \right)}}{\sin{\left(2 t \right)}} - \frac{48 \sin{\left(t \right)} \cos^{2}{\left(2 t \right)}}{\sin^{3}{\left(2 t \right)}} + \cos{\left(t \right)} - \frac{68 \cos{\left(t \right)} \cos{\left(2 t \right)}}{\sin^{2}{\left(2 t \right)}} - \frac{96 \cos{\left(t \right)} \cos^{3}{\left(2 t \right)}}{\sin^{4}{\left(2 t \right)}}$$

Simplify

3-я производная [src]

                    3                                          2                     
  22*sin(t)   96*cos (2*t)*cos(t)   68*cos(t)*cos(2*t)   48*cos (2*t)*sin(t)         
- --------- - ------------------- - ------------------ - ------------------- + cos(t)
   sin(2*t)           4                    2                     3                   
                   sin (2*t)            sin (2*t)             sin (2*t)

$$- \frac{22 \sin{\left(t \right)}}{\sin{\left(2 t \right)}} - \frac{48 \sin{\left(t \right)} \cos^{2}{\left(2 t \right)}}{\sin^{3}{\left(2 t \right)}} + \cos{\left(t \right)} - \frac{68 \cos{\left(t \right)} \cos{\left(2 t \right)}}{\sin^{2}{\left(2 t \right)}} - \frac{96 \cos{\left(t \right)} \cos^{3}{\left(2 t \right)}}{\sin^{4}{\left(2 t \right)}}$$

Simplify

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Derivative of 2cost/sin2t-sint

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