Derivative of cost/(-2sint)

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d t} f{\left(t \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
To find $\frac{d}{d t} g{\left(t \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of sine is cosine:
    
    $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
  So, the result is: $- 2 \cos{\left(t \right)}$
Now plug in to the quotient rule:

$\frac{2 \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}}{4 \sin^{2}{\left(t \right)}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

    2                      
 cos (t)      -1           
--------- - --------*sin(t)
     2      2*sin(t)       
2*sin (t)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of cost/(-2sint)

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The solution