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

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

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

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The solution

You have entered [src] 
cos(x)
------
x + 6
cos⁡(x)x+6\frac{\cos{\left(x \right)}}{x + 6} 
cos(x)/(x + 6)
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+6g{\left(x \right)} = x + 6.

    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+6x + 6 term by term:

      1. The derivative of the constant 66 is zero.

      2. Apply the power rule: xx goes to 11

      The result is: 11

    Now plug in to the quotient rule:

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-1000500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  sin(x)    cos(x) 
- ------ - --------
  x + 6           2
           (x + 6)
−sin⁡(x)x+6−cos⁡(x)(x+6)2- \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
−cos⁡(x)+2sin⁡(x)x+6+2cos⁡(x)(x+6)2x+6\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
sin⁡(x)+3cos⁡(x)x+6−6sin⁡(x)(x+6)2−6cos⁡(x)(x+6)3x+6\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
Derivative of cos(x)/(x+6)
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