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Derivative of 2^x*x^2
Derivative of x^(x^2+1)
Derivative of loga^x
Derivative of cos(x)/(x+6)
Identical expressions
cos(x)/(x+ six)
co sinus of e of (x) divide by (x plus 6)
co sinus of e of (x) divide by (x plus six)
cosx/x+6
cos(x) divide by (x+6)
Similar expressions
cos(x)/(x-6)
cosx/(x+6)

The derivative of the function/ cos(x)/(x+6)

Derivative of cos(x)/(x+6)

The solution

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cos(x)
------
x + 6

$$\frac{\cos{\left(x \right)}}{x + 6}$$

cos(x)/(x + 6)

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 $g{\left(x \right)} = x + 6$ .

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 $x + 6$ term by term:
  1. The derivative of the constant $6$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{- \left(x + 6\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{\left(x + 6\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  sin(x)    cos(x) 
- ------ - --------
  x + 6           2
           (x + 6)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  6*cos(x)   6*sin(x)   3*cos(x)         
- -------- - -------- + -------- + sin(x)
         3          2    6 + x           
  (6 + x)    (6 + x)                     
-----------------------------------------
                  6 + x

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

Simplify

The graph

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Derivative of cos(x)/(x+6)

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The solution