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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /       /      2      \            \
   |     4*\-1 + x  + 2*x/   4*(1 + x)|
12*|-1 - ----------------- + ---------|
   |                  2       1 + 2*x |
   \         (1 + 2*x)                /
---------------------------------------
                        2              
               (1 + 2*x)

$$\frac{12 \left(\frac{4 \left(x + 1\right)}{2 x + 1} - 1 - \frac{4 \left(x^{2} + 2 x - 1\right)}{\left(2 x + 1\right)^{2}}\right)}{\left(2 x + 1\right)^{2}}$$

Simplify

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

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Piecewise:

The solution