The equation presented is:

$y = \frac{2x - 1}{3x + 4}$

We are asked to find the derivative of $y$ with respect to $x$, i.e., $\frac{dy}{dx}$.

Since this is a rational function, we can apply the quotient rule. The quotient rule states that for a function $y = \frac{u(x)}{v(x)}$, the derivative is given by:

$\frac{dy}{dx} = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}$

Steps:

1. Let $u(x) = 2x - 1$ and $v(x) = 3x + 4$.
2. Compute $u'(x)$ and $v'(x)$:
   • $u'(x) = 2$
   • $v'(x) = 3$
3. Apply the quotient rule:

$\frac{dy}{dx} = \frac{(3x + 4) \cdot 2 - (2x - 1) \cdot 3}{(3x + 4)^2}$

Now, I'll simplify the expression.The derivative of the given function $y = \frac{2x - 1}{3x + 4}$ is:

$\frac{dy}{dx} = \frac{11}{(3x + 4)^2}$

Would you like a more detailed explanation of how the quotient rule was applied, or do you have any other questions?

Here are 5 related questions to expand your understanding:

1. How does the quotient rule differ from the product rule in differentiation?
2. What is the physical interpretation of a derivative in real-world applications?
3. How would you apply the chain rule if one of the functions was more complex?
4. Can you find the second derivative of this function?
5. How do you identify when to use the quotient rule over other differentiation techniques?

Tip: When dealing with complex fractions, simplifying expressions before differentiating can save time and reduce errors.