To find the derivative of the function $y = \left(\frac{2x - 3}{5x + 1}\right)^4$, we can use the chain rule combined with the quotient rule.

Step 1: Apply the Chain Rule

Let $u = \frac{2x - 3}{5x + 1}$. Then, $y = u^4$.

The derivative of $y$ with respect to $x$ is:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

The derivative of $y = u^4$ with respect to $u$ is:

$\frac{dy}{du} = 4u^3$

So,

$\frac{dy}{dx} = 4u^3 \cdot \frac{du}{dx}$

Step 2: Differentiate $u = \frac{2x - 3}{5x + 1}$ using the Quotient Rule

The quotient rule is given by:

$\frac{du}{dx} = \frac{(v \cdot \frac{d}{dx}u - u \cdot \frac{d}{dx}v)}{v^2}$

Where $u = 2x - 3$ and $v = 5x + 1$.

Differentiate $u$ and $v$ with respect to $x$:

$\frac{d}{dx}(2x - 3) = 2$ $\frac{d}{dx}(5x + 1) = 5$

Now apply the quotient rule:

$\frac{du}{dx} = \frac{(5x + 1) \cdot 2 - (2x - 3) \cdot 5}{(5x + 1)^2}$

Simplify the numerator:

$\frac{du}{dx} = \frac{10x + 2 - (10x - 15)}{(5x + 1)^2} = \frac{10x + 2 - 10x + 15}{(5x + 1)^2} = \frac{17}{(5x + 1)^2}$

Step 3: Combine the Results

Substitute $u$ back into the derivative:

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

Simplify the final expression:

$\frac{dy}{dx} = \frac{68(2x - 3)^3}{(5x + 1)^5}$

This is the derivative of the given function.

Would you like more details or have any questions?

Relative Questions:

1. How do you apply the chain rule to more complex functions?
2. What are the steps to simplify expressions after applying the quotient rule?
3. How does the power rule work when combined with other differentiation rules?
4. Can you explain how the chain rule and product rule differ?
5. How do you approach differentiating implicit functions?

Tip:

Always check the algebraic simplification carefully to avoid errors, especially after applying the chain rule or quotient rule.