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.

Learn how to find the derivative of the function (6x³-5x²+1)/3x² using calculus techniques. This step-by-step solution includes simplifying the expression and applying the power rule. Ideal for students in Grades 10-12.

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