You entered:

y=(-sint/sin2t)

What you mean?

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

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

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

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

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

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

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

Derivative of y=(-sint/sin2t)

The solution

You have entered [src]

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

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

d /-sin(t) \
--|--------|
dt\sin(2*t)/

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

Transform

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 $g{\left(t \right)} = \sin{\left(2 t \right)}$ .

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 sine is cosine:
    
    $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
  So, the result is: $- \cos{\left(t \right)}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                      /         2     \                         
                                                      |    6*cos (2*t)|                         
                                                    8*|5 + -----------|*cos(2*t)*sin(t)         
     /         2     \                                |        2      |                         
     |    2*cos (2*t)|          6*cos(2*t)*sin(t)     \     sin (2*t) /                         
- 12*|1 + -----------|*cos(t) - ----------------- + ----------------------------------- + cos(t)
     |        2      |               sin(2*t)                     sin(2*t)                      
     \     sin (2*t) /                                                                          
------------------------------------------------------------------------------------------------
                                            sin(2*t)

$$\frac{- 12 \cdot \left(1 + \frac{2 \cos^{2}{\left(2 t \right)}}{\sin^{2}{\left(2 t \right)}}\right) \cos{\left(t \right)} + \frac{8 \cdot \left(5 + \frac{6 \cos^{2}{\left(2 t \right)}}{\sin^{2}{\left(2 t \right)}}\right) \sin{\left(t \right)} \cos{\left(2 t \right)}}{\sin{\left(2 t \right)}} - \frac{6 \sin{\left(t \right)} \cos{\left(2 t \right)}}{\sin{\left(2 t \right)}} + \cos{\left(t \right)}}{\sin{\left(2 t \right)}}$$

Simplify

The graph

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What you mean?

Derivative of y=(-sint/sin2t)

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The solution