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

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cos(x)
------
x + 1

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

cos(x)/(x + 1)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

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

$$\frac{- \cos{\left(x \right)} + \frac{6 \sin{\left(x \right)}}{x + 1} + \frac{30 \cos{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{120 \sin{\left(x \right)}}{\left(x + 1\right)^{3}} - \frac{360 \cos{\left(x \right)}}{\left(x + 1\right)^{4}} + \frac{720 \sin{\left(x \right)}}{\left(x + 1\right)^{5}} + \frac{720 \cos{\left(x \right)}}{\left(x + 1\right)^{6}}}{x + 1}$$

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

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

Simplify

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

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The solution