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y=(x^ two)/(2x- three)
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y=x^2/2x-3
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y=(x^2)/(2x+3)

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

Derivative of y=(x^2)/(2x-3)

The solution

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    2  
   x   
-------
2*x - 3

$$\frac{x^{2}}{2 x - 3}$$

x^2/(2*x - 3)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $2 x - 3$ term by term:
  1. The derivative of the constant $-3$ 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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /            2              \
   |         4*x         4*x   |
12*|-1 - ----------- + --------|
   |               2   -3 + 2*x|
   \     (-3 + 2*x)            /
--------------------------------
                    2           
          (-3 + 2*x)

$$\frac{12 \left(- \frac{4 x^{2}}{\left(2 x - 3\right)^{2}} + \frac{4 x}{2 x - 3} - 1\right)}{\left(2 x - 3\right)^{2}}$$

Simplify

The graph

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Derivative of y=(x^2)/(2x-3)

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