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Identical expressions
(4x^ two +11x- two)/(x+ three)
(4x squared plus 11x minus 2) divide by (x plus 3)
(4x to the power of two plus 11x minus two) divide by (x plus three)
(4x2+11x-2)/(x+3)
4x2+11x-2/x+3
(4x²+11x-2)/(x+3)
(4x to the power of 2+11x-2)/(x+3)
4x^2+11x-2/x+3
(4x^2+11x-2) divide by (x+3)
Similar expressions
(4x^2+11x+2)/(x+3)
(4x^2-11x-2)/(x+3)
(4x^2+11x-2)/(x-3)

The derivative of the function/ (4x^2+11x-2)/(x+3)

Derivative of (4x^2+11x-2)/(x+3)

The solution

You have entered [src]

   2           
4*x  + 11*x - 2
---------------
     x + 3

$$\frac{\left(4 x^{2} + 11 x\right) - 2}{x + 3}$$

(4*x^2 + 11*x - 2)/(x + 3)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 4 x^{2} + 11 x - 2$ and $g{\left(x \right)} = x + 3$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $4 x^{2} + 11 x - 2$ term by term:
  1. The derivative of the constant $-2$ 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: $x^{2}$ goes to $2 x$
    So, the result is: $8 x$
  3. 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: $11$
  The result is: $8 x + 11$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 3$ term by term:
  1. The derivative of the constant $3$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{- 4 x^{2} - 11 x + \left(x + 3\right) \left(8 x + 11\right) + 2}{\left(x + 3\right)^{2}}$
Now simplify:

$\frac{4 x^{2} + 24 x + 35}{x^{2} + 6 x + 9}$

The answer is:

$\frac{4 x^{2} + 24 x + 35}{x^{2} + 6 x + 9}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              2           
11 + 8*x   4*x  + 11*x - 2
-------- - ---------------
 x + 3                2   
               (x + 3)

$$\frac{8 x + 11}{x + 3} - \frac{\left(4 x^{2} + 11 x\right) - 2}{\left(x + 3\right)^{2}}$$

Simplify

The second derivative [src]

  /            2                  \
  |    -2 + 4*x  + 11*x   11 + 8*x|
2*|4 + ---------------- - --------|
  |               2        3 + x  |
  \        (3 + x)                /
-----------------------------------
               3 + x

$$\frac{2 \left(4 - \frac{8 x + 11}{x + 3} + \frac{4 x^{2} + 11 x - 2}{\left(x + 3\right)^{2}}\right)}{x + 3}$$

Simplify

The third derivative [src]

  /                        2       \
  |     11 + 8*x   -2 + 4*x  + 11*x|
6*|-4 + -------- - ----------------|
  |      3 + x                2    |
  \                    (3 + x)     /
------------------------------------
                     2              
              (3 + x)

$$\frac{6 \left(-4 + \frac{8 x + 11}{x + 3} - \frac{4 x^{2} + 11 x - 2}{\left(x + 3\right)^{2}}\right)}{\left(x + 3\right)^{2}}$$

Simplify

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Derivative of (4x^2+11x-2)/(x+3)

The graph:

Piecewise:

The solution