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Similar expressions
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The derivative of the function/ y=sin(t)/(1+2cos(t))

Derivative of y=sin(t)/(1+2cos(t))

The solution

You have entered [src]

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

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

sin(t)/(1 + 2*cos(t))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(t \right)} = \sin{\left(t \right)}$ and $g{\left(t \right)} = 2 \cos{\left(t \right)} + 1$ .

To find $\frac{d}{d t} f{\left(t \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
To find $\frac{d}{d t} g{\left(t \right)}$ :
1. Differentiate $2 \cos{\left(t \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. 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: $- 2 \sin{\left(t \right)}$
  The result is: $- 2 \sin{\left(t \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/       /      2              \               \       
|       | 4*sin (t)           |               |       
|     2*|------------ + cos(t)|               |       
|       \1 + 2*cos(t)         /     4*cos(t)  |       
|-1 + ------------------------- + ------------|*sin(t)
\            1 + 2*cos(t)         1 + 2*cos(t)/       
------------------------------------------------------
                     1 + 2*cos(t)

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

Simplify

The third derivative [src]

                                   /                             2     \                                   
                              2    |      12*cos(t)        24*sin (t)  |     /      2              \       
                         2*sin (t)*|-1 + ------------ + ---------------|     | 4*sin (t)           |       
                2                  |     1 + 2*cos(t)                 2|   6*|------------ + cos(t)|*cos(t)
           6*sin (t)               \                    (1 + 2*cos(t)) /     \1 + 2*cos(t)         /       
-cos(t) - ------------ + ----------------------------------------------- + --------------------------------
          1 + 2*cos(t)                     1 + 2*cos(t)                              1 + 2*cos(t)          
-----------------------------------------------------------------------------------------------------------
                                                1 + 2*cos(t)

$$\frac{\frac{6 \left(\cos{\left(t \right)} + \frac{4 \sin^{2}{\left(t \right)}}{2 \cos{\left(t \right)} + 1}\right) \cos{\left(t \right)}}{2 \cos{\left(t \right)} + 1} - \cos{\left(t \right)} + \frac{2 \left(-1 + \frac{12 \cos{\left(t \right)}}{2 \cos{\left(t \right)} + 1} + \frac{24 \sin^{2}{\left(t \right)}}{\left(2 \cos{\left(t \right)} + 1\right)^{2}}\right) \sin^{2}{\left(t \right)}}{2 \cos{\left(t \right)} + 1} - \frac{6 \sin^{2}{\left(t \right)}}{2 \cos{\left(t \right)} + 1}}{2 \cos{\left(t \right)} + 1}$$

Simplify

The graph

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Derivative of y=sin(t)/(1+2cos(t))

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The solution