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

Derivative of ln(x+1)/x^2

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

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The solution

You have entered [src] 
log(x + 1)
----------
     2    
    x
log⁡(x+1)x2\frac{\log{\left(x + 1 \right)}}{x^{2}} 
log(x + 1)/x^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+1)f{\left(x \right)} = \log{\left(x + 1 \right)} and g(x)=x2g{\left(x \right)} = x^{2}.

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

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

    2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{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. Apply the power rule: xx goes to 11

        2. The derivative of the constant 11 is zero.

        The result is: 11

      The result of the chain rule is:

      1x+1\frac{1}{x + 1}

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

    1. Apply the power rule: x2x^{2} goes to 2x2 x

    Now plug in to the quotient rule:

    x2x+1−2xlog⁡(x+1)x4\frac{\frac{x^{2}}{x + 1} - 2 x \log{\left(x + 1 \right)}}{x^{4}}

  2. Now simplify:

    x−2(x+1)log⁡(x+1)x3(x+1)\frac{x - 2 \left(x + 1\right) \log{\left(x + 1 \right)}}{x^{3} \left(x + 1\right)}

The answer is:

x−2(x+1)log⁡(x+1)x3(x+1)\frac{x - 2 \left(x + 1\right) \log{\left(x + 1 \right)}}{x^{3} \left(x + 1\right)}

The graph
02468-8-6-4-2-1010-100100
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    1        2*log(x + 1)
---------- - ------------
 2                 3     
x *(x + 1)        x
1x2(x+1)−2log⁡(x+1)x3\frac{1}{x^{2} \left(x + 1\right)} - \frac{2 \log{\left(x + 1 \right)}}{x^{3}}
Simplify
The second derivative [src] 
     1           4       6*log(1 + x)
- -------- - --------- + ------------
         2   x*(1 + x)         2     
  (1 + x)                     x      
-------------------------------------
                   2                 
                  x
−1(x+1)2−4x(x+1)+6log⁡(x+1)x2x2\frac{- \frac{1}{\left(x + 1\right)^{2}} - \frac{4}{x \left(x + 1\right)} + \frac{6 \log{\left(x + 1 \right)}}{x^{2}}}{x^{2}}
Simplify
The third derivative [src] 
  /   1       12*log(1 + x)       3            9     \
2*|-------- - ------------- + ---------- + ----------|
  |       3          3                 2    2        |
  \(1 + x)          x         x*(1 + x)    x *(1 + x)/
------------------------------------------------------
                           2                          
                          x
2(1(x+1)3+3x(x+1)2+9x2(x+1)−12log⁡(x+1)x3)x2\frac{2 \left(\frac{1}{\left(x + 1\right)^{3}} + \frac{3}{x \left(x + 1\right)^{2}} + \frac{9}{x^{2} \left(x + 1\right)} - \frac{12 \log{\left(x + 1 \right)}}{x^{3}}\right)}{x^{2}}
Simplify
The graph
Derivative of ln(x+1)/x^2
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