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

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

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

The graph:

from to

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The solution

You have entered [src] 
cos(x)
------
x + 1
cos⁡(x)x+1\frac{\cos{\left(x \right)}}{x + 1} 
cos(x)/(x + 1)
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)=cos⁡(x)f{\left(x \right)} = \cos{\left(x \right)} and g(x)=x+1g{\left(x \right)} = x + 1.

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

    1. The derivative of cosine is negative sine:

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

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

    1. Differentiate x+1x + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. Apply the power rule: xx goes to 11

      The result is: 11

    Now plug in to the quotient rule:

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

  2. Now simplify:

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

The answer is:

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

The graph
0.501.400.600.700.800.901.001.101.201.301-1
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  sin(x)    cos(x) 
- ------ - --------
  x + 1           2
           (x + 1)
−sin⁡(x)x+1−cos⁡(x)(x+1)2- \frac{\sin{\left(x \right)}}{x + 1} - \frac{\cos{\left(x \right)}}{\left(x + 1\right)^{2}}
Simplify
The second derivative [src] 
          2*sin(x)   2*cos(x)
-cos(x) + -------- + --------
           1 + x            2
                     (1 + x) 
-----------------------------
            1 + x
−cos⁡(x)+2sin⁡(x)x+1+2cos⁡(x)(x+1)2x+1\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1}
Simplify
6-я производная [src] 
          360*cos(x)   120*sin(x)   6*sin(x)   30*cos(x)   720*cos(x)   720*sin(x)
-cos(x) - ---------- - ---------- + -------- + --------- + ---------- + ----------
                  4            3     1 + x             2           6            5 
           (1 + x)      (1 + x)                 (1 + x)     (1 + x)      (1 + x)  
----------------------------------------------------------------------------------
                                      1 + x
−cos⁡(x)+6sin⁡(x)x+1+30cos⁡(x)(x+1)2−120sin⁡(x)(x+1)3−360cos⁡(x)(x+1)4+720sin⁡(x)(x+1)5+720cos⁡(x)(x+1)6x+1\frac{- \cos{\left(x \right)} + \frac{6 \sin{\left(x \right)}}{x + 1} + \frac{30 \cos{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{120 \sin{\left(x \right)}}{\left(x + 1\right)^{3}} - \frac{360 \cos{\left(x \right)}}{\left(x + 1\right)^{4}} + \frac{720 \sin{\left(x \right)}}{\left(x + 1\right)^{5}} + \frac{720 \cos{\left(x \right)}}{\left(x + 1\right)^{6}}}{x + 1}
Simplify
The third derivative [src] 
  6*cos(x)   6*sin(x)   3*cos(x)         
- -------- - -------- + -------- + sin(x)
         3          2    1 + x           
  (1 + x)    (1 + x)                     
-----------------------------------------
                  1 + x
sin⁡(x)+3cos⁡(x)x+1−6sin⁡(x)(x+1)2−6cos⁡(x)(x+1)3x+1\frac{\sin{\left(x \right)} + \frac{3 \cos{\left(x \right)}}{x + 1} - \frac{6 \sin{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{6 \cos{\left(x \right)}}{\left(x + 1\right)^{3}}}{x + 1}
Simplify
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