Derivative of (4cos(t))/(-2sin(t))

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

$$\frac{4 \cos{\left(t \right)}}{\left(-1\right) 2 \sin{\left(t \right)}}$$

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(t \right)} = 4 \cos{\left(t \right)}$ and $g{\left(t \right)} = - 2 \sin{\left(t \right)}$ .

To find $\frac{d}{d t} f{\left(t \right)}$ :
1. The derivative of 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: $- 4 \sin{\left(t \right)}$
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:

$\frac{8 \sin^{2}{\left(t \right)} + 8 \cos^{2}{\left(t \right)}}{4 \sin^{2}{\left(t \right)}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

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

$$- 4 \left(- \frac{1}{2 \sin{\left(t \right)}}\right) \sin{\left(t \right)} + \frac{2 \cos^{2}{\left(t \right)}}{\sin^{2}{\left(t \right)}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (4cos(t))/(-2sin(t))

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