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Integral of d{x}:
(x+2)/(x^2+1)
Identical expressions
(x+ two)/(x^ two + one)
(x plus 2) divide by (x squared plus 1)
(x plus two) divide by (x to the power of two plus one)
(x+2)/(x2+1)
x+2/x2+1
(x+2)/(x²+1)
(x+2)/(x to the power of 2+1)
x+2/x^2+1
(x+2) divide by (x^2+1)
Similar expressions
(x+2)/(x^2-1)
(x-2)/(x^2+1)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x + 2$ term by term:
  1. The derivative of the constant $2$ 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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

The graph:

Piecewise:

The solution