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ln((x^ two +3x+ two)/(x+ one))
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ln((x2+3x+2)/(x+1))
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Similar expressions
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The derivative of the function/ ln((x^2+3x+2)/(x+1))

Derivative of ln((x^2+3x+2)/(x+1))

The solution

You have entered [src]

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

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

log((x^2 + 3*x + 2)/(x + 1))

Detail solution

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

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

$\frac{1}{x + 2}$

The answer is:

$\frac{1}{x + 2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln((x^2+3x+2)/(x+1))

The graph:

Piecewise:

The solution