Derivative of (1-cosx)/(x+4)

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1 - cos(x)
----------
  x + 4

$$\frac{1 - \cos{\left(x \right)}}{x + 4}$$

(1 - cos(x))/(x + 4)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 1 - \cos{\left(x \right)}$ and $g{\left(x \right)} = x + 4$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $1 - \cos{\left(x \right)}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $\sin{\left(x \right)}$
  The result is: $\sin{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 4$ term by term:
  1. The derivative of the constant $4$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{\left(x + 4\right) \sin{\left(x \right)} + \cos{\left(x \right)} - 1}{\left(x + 4\right)^{2}}$

The answer is:

$\frac{\left(x + 4\right) \sin{\left(x \right)} + \cos{\left(x \right)} - 1}{\left(x + 4\right)^{2}}$

The graph

Plot the graph f(x)

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The first derivative [src]

sin(x)   1 - cos(x)
------ - ----------
x + 4            2 
          (x + 4)

$$- \frac{1 - \cos{\left(x \right)}}{\left(x + 4\right)^{2}} + \frac{\sin{\left(x \right)}}{x + 4}$$

Simplify

The second derivative [src]

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

$$\frac{\cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{x + 4} - \frac{2 \left(\cos{\left(x \right)} - 1\right)}{\left(x + 4\right)^{2}}}{x + 4}$$

Simplify

The third derivative [src]

          3*cos(x)   6*(-1 + cos(x))   6*sin(x)
-sin(x) - -------- + --------------- + --------
           4 + x                3             2
                         (4 + x)       (4 + x) 
-----------------------------------------------
                     4 + x

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

Simplify

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Derivative of (1-cosx)/(x+4)

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The solution