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y=sinx/(1-cosx)²

The derivative of the function/ y=sinx/(1+cosx)²

Derivative of y=sinx/(1+cosx)²

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)) /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

$\frac{\left(\cos{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)} + \left(2 \cos{\left(x \right)} + 2\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                       2     
    cos(x)        2*sin (x)  
------------- + -------------
            2               3
(1 + cos(x))    (1 + cos(x))

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

Simplify

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))

$$\frac{\left(-1 + \frac{2 \left(\cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)}{\cos{\left(x \right)} + 1} + \frac{4 \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}$$

Simplify

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))

$$\frac{\frac{2 \left(-1 + \frac{9 \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{12 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}\right) \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1} - \cos{\left(x \right)} + \frac{6 \left(\cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} - \frac{6 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}}{\left(\cos{\left(x \right)} + 1\right)^{2}}$$

Simplify

The graph

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Derivative of y=sinx/(1+cosx)²

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Piecewise:

The solution