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y=sinx/(1+cosx)²

Derivative of y=sinx/(1+cosx)²

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
    sin(x)   
-------------
            2
(1 + cos(x)) 
$$\frac{\sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}$$
d /    sin(x)   \
--|-------------|
dx|            2|
  \(1 + cos(x)) /
$$\frac{d}{d x} \frac{\sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}$$
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of sine is cosine:

    To find :

    1. Let .

    2. Apply the power rule: goes to

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

      1. Differentiate term by term:

        1. The derivative of the constant is zero.

        2. The derivative of cosine is negative sine:

        The result is:

      The result of the chain rule is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                       2     
    cos(x)        2*sin (x)  
------------- + -------------
            2               3
(1 + cos(x))    (1 + cos(x)) 
$$\frac{\cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \frac{2 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{3}}$$
The second derivative [src]
/       /     2             \             \       
|       |3*sin (x)          |             |       
|     2*|---------- + cos(x)|             |       
|       \1 + cos(x)         /    4*cos(x) |       
|-1 + ----------------------- + ----------|*sin(x)
\            1 + cos(x)         1 + cos(x)/       
--------------------------------------------------
                              2                   
                  (1 + cos(x))                    
$$\frac{\left(-1 + \frac{2 \left(\cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)}{\cos{\left(x \right)} + 1} + \frac{4 \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}$$
The third derivative [src]
                                 /                          2    \                                 
                            2    |      9*cos(x)      12*sin (x) |     /     2             \       
                       2*sin (x)*|-1 + ---------- + -------------|     |3*sin (x)          |       
               2                 |     1 + cos(x)               2|   6*|---------- + cos(x)|*cos(x)
          6*sin (x)              \                  (1 + cos(x)) /     \1 + cos(x)         /       
-cos(x) - ---------- + ------------------------------------------- + ------------------------------
          1 + cos(x)                    1 + cos(x)                             1 + cos(x)          
---------------------------------------------------------------------------------------------------
                                                       2                                           
                                           (1 + cos(x))                                            
$$\frac{\frac{2 \left(-1 + \frac{9 \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{12 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}\right) \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1} - \cos{\left(x \right)} + \frac{6 \left(\cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} - \frac{6 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}}{\left(\cos{\left(x \right)} + 1\right)^{2}}$$
The graph
Derivative of y=sinx/(1+cosx)²