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lnx/(x-1)^2

The derivative of the function/ lnx/(x+1)^2

Derivative of lnx/(x+1)^2

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

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

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. 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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

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)

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

Simplify

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Derivative of lnx/(x+1)^2

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Piecewise:

The solution