# Find f'(x) and simplify. f(x) = ((3x - 4) / (2x + 3))

## Question:

Find {eq}f'(x) {/eq} and simplify.

{eq}f(x) =\frac{3x - 4}{2x + 3} {/eq}

## Quotient Rule for Differentiation:

This rule of differentiation is the rule to get the derivative of the fractional or the division expression. This rule is used only when the numerator and the denominator is not the unity.

To find the derivative {eq}f'(x) {/eq} of the function

{eq}f(x) =\frac{3x - 4}{2x + 3} {/eq}

we will use the quotient rule as follows:

{eq}f'(x)= \frac{d}{dx}\left(\frac{3x\:-\:4}{2x\:+\:3}\right)\\ \displaystyle =\frac{\frac{d}{dx}\left(3x-4\right)\left(2x+3\right)-\frac{d}{dx}\left(2x+3\right)\left(3x-4\right)}{\left(2x+3\right)^2}~~~~~~~~~~~~~~\left [ \because \left(\frac{f}{g}\right)'=\frac{f\:'\cdot g-g'\cdot f}{g^2} \right ]\\ =\frac{3\left(2x+3\right)-2\left(3x-4\right)}{\left(2x+3\right)^2}~~~~~~~~~~~~~\left [ \because \frac{d}{dx}\left(x\right)=1 \right ]\\ =\frac{17}{\left(2x+3\right)^2} {/eq}

So this is the simplified result. 