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

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log(x)
------
x + 1

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

d /log(x)\
--|------|
dx\x + 1 /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(x \right)}$ and $g{\left(x \right)} = x + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

2    6*log(x)       3            6     
-- - -------- + ---------- + ----------
 3          3    2                    2
x    (1 + x)    x *(1 + x)   x*(1 + x) 
---------------------------------------
                 1 + x

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

Simplify

The graph

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

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The solution