Derivative of log(x/(x+1))

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

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

log(x/(x + 1))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of log(x/(x+1))

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Piecewise:

The solution