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(3*x+5)/(2*x-1)

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Derivative of:
Derivative of x^12
Derivative of -e^x
Derivative of e^(2-x)
Derivative of x^2-4*x
Identical expressions
(three *x+ five)/(two *x- one)
(3 multiply by x plus 5) divide by (2 multiply by x minus 1)
(three multiply by x plus five) divide by (two multiply by x minus one)
(3x+5)/(2x-1)
3x+5/2x-1
(3*x+5) divide by (2*x-1)
Similar expressions
(3*x+5)/(2*x+1)
f(x)=(-3x+5)/(2x-1)
(3x+5)/(2x-1)
(3*x-5)/(2*x-1)

The derivative of the function/ (3*x+5)/(2*x-1)

Derivative of (3x+5)/(2x-1)

The solution

You have entered [src]

3*x + 5
-------
2*x - 1

$$\frac{3 x + 5}{2 x - 1}$$

(3*x + 5)/(2*x - 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 3 x + 5$ and $g{\left(x \right)} = 2 x - 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 x + 5$ term by term:
  1. The derivative of the constant $5$ 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:

$- \frac{13}{\left(2 x - 1\right)^{2}}$

The answer is:

$- \frac{13}{\left(2 x - 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3      2*(3*x + 5)
------- - -----------
2*x - 1             2
           (2*x - 1)

$$\frac{3}{2 x - 1} - \frac{2 \left(3 x + 5\right)}{\left(2 x - 1\right)^{2}}$$

Simplify

The second derivative [src]

  /     2*(5 + 3*x)\
4*|-3 + -----------|
  \       -1 + 2*x /
--------------------
              2     
    (-1 + 2*x)

$$\frac{4 \left(-3 + \frac{2 \left(3 x + 5\right)}{2 x - 1}\right)}{\left(2 x - 1\right)^{2}}$$

Simplify

The third derivative [src]

   /    2*(5 + 3*x)\
24*|3 - -----------|
   \      -1 + 2*x /
--------------------
              3     
    (-1 + 2*x)

$$\frac{24 \left(3 - \frac{2 \left(3 x + 5\right)}{2 x - 1}\right)}{\left(2 x - 1\right)^{3}}$$

Simplify

The graph

Derivative of (3*x+5)/(2*x-1)