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Derivative of:
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Derivative of x×ln(x)
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Derivative of tg(x)^3
Integral of d{x}:
(2*x+1)/(x+2)
Graphing y =:
(2*x+1)/(x+2)
Identical expressions
(two *x+ one)/(x+ two)
(2 multiply by x plus 1) divide by (x plus 2)
(two multiply by x plus one) divide by (x plus two)
(2x+1)/(x+2)
2x+1/x+2
(2*x+1) divide by (x+2)
Similar expressions
(2*x-1)/(x+2)
(2*x+1)/(x-2)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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