# F(x)= fraction {x^2 - 1}{2x - 3} Find the derivative and the domain

## Question:

{eq}\displaystyle F(x)= \frac {x^2 - 1}{2x - 3 } {/eq} Find the derivative and the domain

## Quotient Rule:

When the function to be differentiated is in the form of a fraction, then quotient rule of differentiation is used.

If the given fraction is N/D, then quotient rule is given by the following formula:

{eq}\dfrac {d}{dx} \dfrac {N}{D} = \dfrac {DN' - N D'}{D^2} {/eq}

{eq}\displaystyle F(x)= \frac {x^2 - 1}{2x - 3 } {/eq}

On differentiating the given function:

{eq}F'(x) = \dfrac { (2x - 3 ) \frac {d}{dx} (x^2 - 1) - (x^2 - 1) \frac {d}{dx}(2x - 3 ) }{(2x - 3 )^2} \\ = \dfrac { (2x - 3 ) (2x) - (x^2 - 1)(2 ) }{(2x - 3 )^2} \\ = \dfrac { 4x^2 - 6x - 2x^2 + 2 }{(2x - 3 )^2} \\ = \dfrac { 2x^2 - 6x + 2 }{(2x - 3 )^2} \\ {/eq}

Domain:

At x= 3/2, the denominator becomes zero and the function becomes undefined. Therefore, except at x=3/2, the function is real and defined.

Hence, the domain interval is {eq}(-\infty, 3/2) \cup (3/2, \infty). {/eq}