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sin(x)/(1+cos(x))
The derivative of the function/ sin(x)/(1+cos(x))

Derivative of sin(x)/(1+cos(x))

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

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The solution

You have entered [src] 
  sin(x)  
----------
1 + cos(x)
sin⁡(x)cos⁡(x)+1\frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1} 
sin(x)/(1 + cos(x))
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)+1g{\left(x \right)} = \cos{\left(x \right)} + 1.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate cos⁡(x)+1\cos{\left(x \right)} + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. The derivative of cosine is negative sine:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

      The result is: −sin⁡(x)- \sin{\left(x \right)}

    Now plug in to the quotient rule:

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-2000020000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                   2      
  cos(x)        sin (x)   
---------- + -------------
1 + cos(x)               2
             (1 + cos(x))
cos⁡(x)cos⁡(x)+1+sin⁡2(x)(cos⁡(x)+1)2\frac{\cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}
Simplify
The second derivative [src] 
/          2                          \       
|     2*sin (x)                       |       
|     ---------- + cos(x)             |       
|     1 + cos(x)             2*cos(x) |       
|-1 + ------------------- + ----------|*sin(x)
\          1 + cos(x)       1 + cos(x)/       
----------------------------------------------
                  1 + cos(x)
(−1+cos⁡(x)+2sin⁡2(x)cos⁡(x)+1cos⁡(x)+1+2cos⁡(x)cos⁡(x)+1)sin⁡(x)cos⁡(x)+1\frac{\left(-1 + \frac{\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}}{\cos{\left(x \right)} + 1} + \frac{2 \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}
Simplify
The third derivative [src] 
                               /                         2     \                                 
                          2    |      6*cos(x)      6*sin (x)  |     /     2             \       
                       sin (x)*|-1 + ---------- + -------------|     |2*sin (x)          |       
               2               |     1 + cos(x)               2|   3*|---------- + cos(x)|*cos(x)
          3*sin (x)            \                  (1 + cos(x)) /     \1 + cos(x)         /       
-cos(x) - ---------- + ----------------------------------------- + ------------------------------
          1 + cos(x)                   1 + cos(x)                            1 + cos(x)          
-------------------------------------------------------------------------------------------------
                                            1 + cos(x)
−cos⁡(x)+3(cos⁡(x)+2sin⁡2(x)cos⁡(x)+1)cos⁡(x)cos⁡(x)+1+(−1+6cos⁡(x)cos⁡(x)+1+6sin⁡2(x)(cos⁡(x)+1)2)sin⁡2(x)cos⁡(x)+1−3sin⁡2(x)cos⁡(x)+1cos⁡(x)+1\frac{- \cos{\left(x \right)} + \frac{3 \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\left(-1 + \frac{6 \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{6 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}\right) \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1} - \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}}{\cos{\left(x \right)} + 1}
Simplify
The graph
Derivative of sin(x)/(1+cos(x))
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