Derivative of (1+lnx)/(1+x)²

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(x \right)} + 1$ and $g{\left(x \right)} = \left(x + 1\right)^{2}$ .

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

$\frac{- \left(2 x + 2\right) \left(\log{\left(x \right)} + 1\right) + \frac{\left(x + 1\right)^{2}}{x}}{\left(x + 1\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1        (1 + log(x))*(-2 - 2*x)
---------- + -----------------------
         2                  4       
x*(1 + x)            (1 + x)

$$\frac{\left(- 2 x - 2\right) \left(\log{\left(x \right)} + 1\right)}{\left(x + 1\right)^{4}} + \frac{1}{x \left(x + 1\right)^{2}}$$

Simplify

The second derivative [src]

  1        4       6*(1 + log(x))
- -- - --------- + --------------
   2   x*(1 + x)             2   
  x                   (1 + x)    
---------------------------------
                    2            
             (1 + x)

$$\frac{\frac{6 \left(\log{\left(x \right)} + 1\right)}{\left(x + 1\right)^{2}} - \frac{4}{x \left(x + 1\right)} - \frac{1}{x^{2}}}{\left(x + 1\right)^{2}}$$

Simplify

The third derivative [src]

  /1    12*(1 + log(x))       3            9     \
2*|-- - --------------- + ---------- + ----------|
  | 3              3       2                    2|
  \x        (1 + x)       x *(1 + x)   x*(1 + x) /
--------------------------------------------------
                            2                     
                     (1 + x)

$$\frac{2 \left(- \frac{12 \left(\log{\left(x \right)} + 1\right)}{\left(x + 1\right)^{3}} + \frac{9}{x \left(x + 1\right)^{2}} + \frac{3}{x^{2} \left(x + 1\right)} + \frac{1}{x^{3}}\right)}{\left(x + 1\right)^{2}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (1+lnx)/(1+x)²

The graph:

Piecewise:

The solution