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

Derivative of lnx/(x+1)^2

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

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The solution

You have entered [src] 
 log(x) 
--------
       2
(x + 1)
log⁡(x)(x+1)2\frac{\log{\left(x \right)}}{\left(x + 1\right)^{2}} 
log(x)/(x + 1)^2
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)} and g(x)=(x+1)2g{\left(x \right)} = \left(x + 1\right)^{2}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=x+1u = x + 1.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddx(x+1)\frac{d}{d x} \left(x + 1\right):

      1. Differentiate x+1x + 1 term by term:

        1. The derivative of the constant 11 is zero.

        2. Apply the power rule: xx goes to 11

        The result is: 11

      The result of the chain rule is:

      2x+22 x + 2

    Now plug in to the quotient rule:

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-101020-10
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)
(−2x−2)log⁡(x)(x+1)4+1x(x+1)2\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)
6log⁡(x)(x+1)2−4x(x+1)−1x2(x+1)2\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)
2(−12log⁡(x)(x+1)3+9x(x+1)2+3x2(x+1)+1x3)(x+1)2\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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