Derivative of (x-1)/(x²+1)

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x - 1 
------
 2    
x  + 1

$$\frac{x - 1}{x^{2} + 1}$$

(x - 1)/(x^2 + 1)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1      2*x*(x - 1)
------ - -----------
 2                2 
x  + 1    / 2    \  
          \x  + 1/

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

Simplify

The second derivative [src]

  /                /         2 \\
  |                |      4*x  ||
2*|-2*x + (-1 + x)*|-1 + ------||
  |                |          2||
  \                \     1 + x //
---------------------------------
                    2            
            /     2\             
            \1 + x /

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

Simplify

The third derivative [src]

  /                           /         2 \\
  |                           |      2*x  ||
  |              4*x*(-1 + x)*|-1 + ------||
  |         2                 |          2||
  |      4*x                  \     1 + x /|
6*|-1 + ------ - --------------------------|
  |          2                  2          |
  \     1 + x              1 + x           /
--------------------------------------------
                         2                  
                 /     2\                   
                 \1 + x /

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

Simplify

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Derivative of (x-1)/(x²+1)

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The solution