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y(x)=(3x-1)/(2x+1)

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Identical expressions
y(x)=(3x- one)/(2x+ one)
y(x) equally (3x minus 1) divide by (2x plus 1)
y(x) equally (3x minus one) divide by (2x plus one)
yx=3x-1/2x+1
y(x)=(3x-1) divide by (2x+1)
Similar expressions
y(x)=(3x+1)/(2x+1)
3*x-1/2*x+1
y(x)=(3x-1)/(2x-1)

The derivative of the function/ y(x)=(3x-1)/(2x+1)

Derivative of y(x)=(3x-1)/(2x+1)

The solution

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 x - 1$ 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: $3$
  The result is: $3$
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:

$\frac{5}{\left(2 x + 1\right)^{2}}$

The answer is:

$\frac{5}{\left(2 x + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /     2*(-1 + 3*x)\
4*|-3 + ------------|
  \       1 + 2*x   /
---------------------
               2     
      (1 + 2*x)

$$\frac{4 \left(-3 + \frac{2 \left(3 x - 1\right)}{2 x + 1}\right)}{\left(2 x + 1\right)^{2}}$$

Simplify

The third derivative [src]

   /    2*(-1 + 3*x)\
24*|3 - ------------|
   \      1 + 2*x   /
---------------------
               3     
      (1 + 2*x)

$$\frac{24 \left(3 - \frac{2 \left(3 x - 1\right)}{2 x + 1}\right)}{\left(2 x + 1\right)^{3}}$$

Simplify

The graph

Derivative of y(x)=(3x-1)/(2x+1)