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

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

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

log((x + 1)/(x + 2))

Detail solution

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

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

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