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Integral of d{x}:
log(x+1)/x^2
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Similar expressions
log(x-1)/x^2

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

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

The solution

You have entered [src]

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

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

log(x + 1)/x^2

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

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

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

Simplify

The second derivative [src]

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

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

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

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

Simplify

The graph

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

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The solution