To find the derivative $\frac{dy}{dx}$ of the function $\frac{6x^3 - 5x^2 + 1}{3x^2}$, we can simplify the function first and then differentiate it.

Step 1: Simplify the function

$\frac{6x^3 - 5x^2 + 1}{3x^2}$

We can simplify this by dividing each term in the numerator by $3x^2$:

$\frac{6x^3}{3x^2} - \frac{5x^2}{3x^2} + \frac{1}{3x^2} = 2x - \frac{5}{3} + \frac{1}{3x^2}$

Step 2: Differentiate the simplified function

Now we differentiate each term with respect to $x$:

1. The derivative of $2x$ is $2$.
2. The derivative of $-\frac{5}{3}$ is $0$ (since it's a constant).
3. The derivative of $\frac{1}{3x^2}$ is $\frac{1}{3} \cdot (-2x^{-3}) = -\frac{2}{3x^3}$.

So, the derivative is: $\frac{dy}{dx} = 2 - \frac{2}{3x^3}$

Final Answer:

$\frac{dy}{dx} = 2 - \frac{2}{3x^3}$

Would you like more details or have any questions about this process?

Here are five related questions you might find useful:

1. How would you find the second derivative of the function?
2. What is the derivative of $\frac{1}{x^n}$ for any integer $n$?
3. How can you apply the chain rule to more complex functions?
4. What happens to the derivative as $x$ approaches zero?
5. How would you integrate the original function?

Tip: When simplifying fractions involving polynomials, divide each term in the numerator by the denominator to make differentiation easier.