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Similar expressions
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The derivative of the function/ (3+x^2)/(x+1)

Derivative of (3+x^2)/(x+1)

The solution

You have entered [src]

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

$$\frac{x^{2} + 3}{x + 1}$$

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{2} + 3$ and $g{\left(x \right)} = x + 1$ .

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        2         
   3 + x      2*x 
- -------- + -----
         2   x + 1
  (x + 1)

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

Simplify

The second derivative [src]

  /          2         \
  |     3 + x      2*x |
2*|1 + -------- - -----|
  |           2   1 + x|
  \    (1 + x)         /
------------------------
         1 + x

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

Simplify

The third derivative [src]

  /           2         \
  |      3 + x      2*x |
6*|-1 - -------- + -----|
  |            2   1 + x|
  \     (1 + x)         /
-------------------------
                2        
         (1 + x)

$$\frac{6 \left(\frac{2 x}{x + 1} - 1 - \frac{x^{2} + 3}{\left(x + 1\right)^{2}}\right)}{\left(x + 1\right)^{2}}$$

Simplify

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Derivative of (3+x^2)/(x+1)

The graph:

Piecewise:

The solution