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Identical expressions
y=sin(6x)/(one +cos6x)
y equally sinus of (6x) divide by (1 plus co sinus of e of 6x)
y equally sinus of (6x) divide by (one plus co sinus of e of 6x)
y=sin6x/1+cos6x
y=sin(6x) divide by (1+cos6x)
Similar expressions
y=sin(6x)/(1-cos6x)

The derivative of the function/ y=sin(6x)/(1+cos6x)

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

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

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(6 x \right)}$ and $g{\left(x \right)} = \cos{\left(6 x \right)} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 6 x$ .
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 $\frac{d}{d x} 6 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $6$
  The result of the chain rule is:
  
  $6 \cos{\left(6 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\cos{\left(6 x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = 6 x$ .
  3. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} 6 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $6$
    The result of the chain rule is:
    
    $- 6 \sin{\left(6 x \right)}$
  The result is: $- 6 \sin{\left(6 x \right)}$
Now plug in to the quotient rule:

$\frac{6 \left(\cos{\left(6 x \right)} + 1\right) \cos{\left(6 x \right)} + 6 \sin^{2}{\left(6 x \right)}}{\left(\cos{\left(6 x \right)} + 1\right)^{2}}$
Now simplify:

$\frac{6}{\cos{\left(6 x \right)} + 1}$

The answer is:

$\frac{6}{\cos{\left(6 x \right)} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      2       
 6*cos(6*x)      6*sin (6*x)  
------------ + ---------------
1 + cos(6*x)                 2
               (1 + cos(6*x))

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

Simplify

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)

$$\frac{36 \left(-1 + \frac{\cos{\left(6 x \right)} + \frac{2 \sin^{2}{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}}{\cos{\left(6 x \right)} + 1} + \frac{2 \cos{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}\right) \sin{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}$$

Simplify

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)

$$\frac{216 \left(- \cos{\left(6 x \right)} + \frac{3 \left(\cos{\left(6 x \right)} + \frac{2 \sin^{2}{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}\right) \cos{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1} + \frac{\left(-1 + \frac{6 \cos{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1} + \frac{6 \sin^{2}{\left(6 x \right)}}{\left(\cos{\left(6 x \right)} + 1\right)^{2}}\right) \sin^{2}{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1} - \frac{3 \sin^{2}{\left(6 x \right)}}{\cos{\left(6 x \right)} + 1}\right)}{\cos{\left(6 x \right)} + 1}$$

Simplify

The graph

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

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The solution