Derivative of (ln(x+8)-3x+2)/x

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log(x + 8) - 3*x + 2
--------------------
         x

$$\frac{\left(- 3 x + \log{\left(x + 8 \right)}\right) + 2}{x}$$

(log(x + 8) - 3*x + 2)/x

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = - 3 x + \log{\left(x + 8 \right)} + 2$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $- 3 x + \log{\left(x + 8 \right)} + 2$ term by term:
  1. The derivative of the constant $2$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $-3$
  3. Let $u = x + 8$ .
  4. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  5. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 8\right)$ :
    1. Differentiate $x + 8$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $8$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $\frac{1}{x + 8}$
  The result is: $-3 + \frac{1}{x + 8}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

$\frac{x \left(-3 + \frac{1}{x + 8}\right) + 3 x - \log{\left(x + 8 \right)} - 2}{x^{2}}$
Now simplify:

$- \frac{x \log{\left(x + 8 \right)} + x + 8 \log{\left(x + 8 \right)} + 16}{x^{2} \left(x + 8\right)}$

The answer is:

$- \frac{x \log{\left(x + 8 \right)} + x + 8 \log{\left(x + 8 \right)} + 16}{x^{2} \left(x + 8\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       1                         
-3 + -----                       
     x + 8   log(x + 8) - 3*x + 2
---------- - --------------------
    x                  2         
                      x

$$\frac{-3 + \frac{1}{x + 8}}{x} - \frac{\left(- 3 x + \log{\left(x + 8 \right)}\right) + 2}{x^{2}}$$

Simplify

The second derivative [src]

               /      1  \                           
             2*|3 - -----|                           
     1         \    8 + x/   2*(2 - 3*x + log(8 + x))
- -------- + ------------- + ------------------------
         2         x                     2           
  (8 + x)                               x            
-----------------------------------------------------
                          x

$$\frac{- \frac{1}{\left(x + 8\right)^{2}} + \frac{2 \left(3 - \frac{1}{x + 8}\right)}{x} + \frac{2 \left(- 3 x + \log{\left(x + 8 \right)} + 2\right)}{x^{2}}}{x}$$

Simplify

The third derivative [src]

                                        /      1  \             
                                      6*|3 - -----|             
   2       6*(2 - 3*x + log(8 + x))     \    8 + x/       3     
-------- - ------------------------ - ------------- + ----------
       3               3                     2                 2
(8 + x)               x                     x         x*(8 + x) 
----------------------------------------------------------------
                               x

$$\frac{\frac{2}{\left(x + 8\right)^{3}} + \frac{3}{x \left(x + 8\right)^{2}} - \frac{6 \left(3 - \frac{1}{x + 8}\right)}{x^{2}} - \frac{6 \left(- 3 x + \log{\left(x + 8 \right)} + 2\right)}{x^{3}}}{x}$$

Simplify

The graph

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Derivative of (ln(x+8)-3x+2)/x

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The solution