# 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.

## Answer and Explanation:

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.

#### Learn more about this topic:

When to Use the Quotient Rule for Differentiation

from Math 104: Calculus

Chapter 8 / Lesson 8
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