Derivative of (ln(x+3)/x+3)/(3+x)

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log(x + 3)    
---------- + 3
    x         
--------------
    3 + x

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

(log(x + 3)/x + 3)/(3 + x)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

$\frac{x \left(3 x + 10\right) - \left(2 x + 3\right) \left(3 x + \log{\left(x + 3 \right)}\right)}{x^{2} \left(x + 3\right)^{2}}$

The answer is:

$\frac{x \left(3 x + 10\right) - \left(2 x + 3\right) \left(3 x + \log{\left(x + 3 \right)}\right)}{x^{2} \left(x + 3\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

3-я производная [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (ln(x+3)/x+3)/(3+x)

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