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Derivative of lnx/(x+1)^2

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
 log(x) 
--------
       2
(x + 1) 
$$\frac{\log{\left(x \right)}}{\left(x + 1\right)^{2}}$$
log(x)/(x + 1)^2
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of is .

    To find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate term by term:

        1. The derivative of the constant is zero.

        2. Apply the power rule: goes to

        The result is:

      The result of the chain rule is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
    1        (-2 - 2*x)*log(x)
---------- + -----------------
         2               4    
x*(x + 1)         (x + 1)     
$$\frac{\left(- 2 x - 2\right) \log{\left(x \right)}}{\left(x + 1\right)^{4}} + \frac{1}{x \left(x + 1\right)^{2}}$$
The second derivative [src]
  1        4       6*log(x)
- -- - --------- + --------
   2   x*(1 + x)          2
  x                (1 + x) 
---------------------------
                 2         
          (1 + x)          
$$\frac{\frac{6 \log{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{4}{x \left(x + 1\right)} - \frac{1}{x^{2}}}{\left(x + 1\right)^{2}}$$
The third derivative [src]
  /1    12*log(x)       3            9     \
2*|-- - --------- + ---------- + ----------|
  | 3           3    2                    2|
  \x     (1 + x)    x *(1 + x)   x*(1 + x) /
--------------------------------------------
                         2                  
                  (1 + x)                   
$$\frac{2 \left(- \frac{12 \log{\left(x \right)}}{\left(x + 1\right)^{3}} + \frac{9}{x \left(x + 1\right)^{2}} + \frac{3}{x^{2} \left(x + 1\right)} + \frac{1}{x^{3}}\right)}{\left(x + 1\right)^{2}}$$