To solve the given problem, we need to find the derivative of the function

$y = \left(\frac{-5x + 4}{-3x + 1}\right)^3$

Step-by-Step Solution:

We are dealing with a composition of functions here, so we will apply the chain rule and quotient rule.

1. Apply the chain rule: $\frac{dy}{dx} = 3 \left( \frac{-5x + 4}{-3x + 1} \right)^2 \cdot \frac{d}{dx}\left( \frac{-5x + 4}{-3x + 1} \right)$ So, we first take the derivative of the outer function and leave the inner function intact, multiplying by the derivative of the inner function (the quotient).

2. Apply the quotient rule for the inner derivative $\frac{-5x + 4}{-3x + 1}$:

   The quotient rule states that for $\frac{f(x)}{g(x)}$, the derivative is: $\frac{d}{dx}\left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ Here:

   • $f(x) = -5x + 4$ and $g(x) = -3x + 1$
   • $f'(x) = -5$ and $g'(x) = -3$

   Applying the quotient rule: $\frac{d}{dx}\left( \frac{-5x + 4}{-3x + 1} \right) = \frac{(-5)(-3x + 1) - (-5x + 4)(-3)}{(-3x + 1)^2}$ Simplifying: $= \frac{15x - 5 - (-15x + 12)}{(-3x + 1)^2}$ $= \frac{15x - 5 + 15x - 12}{(-3x + 1)^2}$ $= \frac{30x - 17}{(-3x + 1)^2}$

3. Combine everything: Now substitute this back into the chain rule expression: $\frac{dy}{dx} = 3 \left( \frac{-5x + 4}{-3x + 1} \right)^2 \cdot \frac{30x - 17}{(-3x + 1)^2}$ Simplifying: $\frac{dy}{dx} = \frac{3(-5x + 4)^2 (30x - 17)}{(-3x + 1)^4}$

So, the derivative of the given function in factored form is:

$\frac{dy}{dx} = \frac{3(-5x + 4)^2 (30x - 17)}{(-3x + 1)^4}$

Let me know if you'd like further clarification on any step!

Related questions:

1. How does the chain rule apply to functions with multiple layers of composition?
2. What is the quotient rule, and why is it necessary for rational functions?
3. Can you explain how factoring impacts simplifying derivatives?
4. How would the solution change if the exponent on the original function were different (e.g., $n = 4$)?
5. How can we apply the product rule alongside the chain and quotient rules in more complex cases?

Tip:

Always simplify the inner function first when applying the quotient rule to avoid excessive complexity later in the problem.