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y=sin(6x)/(1+cos6x)

Derivative of y=sin(6x)/(1+cos6x)

Function f() - derivative -N order at the point
v

The graph:

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The solution

You have entered [src]
  sin(6*x)  
------------
1 + cos(6*x)
$$\frac{\sin{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}$$
sin(6*x)/(1 + cos(6*x))
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Let .

    2. The derivative of sine is cosine:

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. Let .

      3. The derivative of cosine is negative sine:

      4. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                      2       
 6*cos(6*x)      6*sin (6*x)  
------------ + ---------------
1 + cos(6*x)                 2
               (1 + cos(6*x)) 
$$\frac{6 \cos{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1} + \frac{6 \sin^{2}{\left(6 x \right)}}{\left(\cos{\left(6 x \right)} + 1\right)^{2}}$$
The second derivative [src]
   /          2                                \         
   |     2*sin (6*x)                           |         
   |     ------------ + cos(6*x)               |         
   |     1 + cos(6*x)               2*cos(6*x) |         
36*|-1 + ----------------------- + ------------|*sin(6*x)
   \           1 + cos(6*x)        1 + cos(6*x)/         
---------------------------------------------------------
                       1 + cos(6*x)                      
$$\frac{36 \left(-1 + \frac{\cos{\left(6 x \right)} + \frac{2 \sin^{2}{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}}{\cos{\left(6 x \right)} + 1} + \frac{2 \cos{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}\right) \sin{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}$$
The third derivative [src]
    /                                     /                           2       \                                       \
    |                              2      |      6*cos(6*x)      6*sin (6*x)  |     /     2                 \         |
    |                           sin (6*x)*|-1 + ------------ + ---------------|     |2*sin (6*x)            |         |
    |                 2                   |     1 + cos(6*x)                 2|   3*|------------ + cos(6*x)|*cos(6*x)|
    |            3*sin (6*x)              \                    (1 + cos(6*x)) /     \1 + cos(6*x)           /         |
216*|-cos(6*x) - ------------ + ----------------------------------------------- + ------------------------------------|
    \            1 + cos(6*x)                     1 + cos(6*x)                                1 + cos(6*x)            /
-----------------------------------------------------------------------------------------------------------------------
                                                      1 + cos(6*x)                                                     
$$\frac{216 \left(- \cos{\left(6 x \right)} + \frac{3 \left(\cos{\left(6 x \right)} + \frac{2 \sin^{2}{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}\right) \cos{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1} + \frac{\left(-1 + \frac{6 \cos{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1} + \frac{6 \sin^{2}{\left(6 x \right)}}{\left(\cos{\left(6 x \right)} + 1\right)^{2}}\right) \sin^{2}{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1} - \frac{3 \sin^{2}{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}\right)}{\cos{\left(6 x \right)} + 1}$$
The graph
Derivative of y=sin(6x)/(1+cos6x)