To find the derivative \( \frac{dy}{dx} \) of the function \( y = 3 - \frac{5}{x^2} + \frac{2x^{1/2}}{5} \), we'll apply the following rules:

1. **Constant Rule**: The derivative of a constant is 0.
2. **Power Rule**: For \( x^n \), the derivative is \( nx^{n-1} \).
3. **Quotient Rule** for powers of \( x \): \( \frac{d}{dx}\left(\frac{a}{x^n}\right) = -a n x^{-(n+1)} \).

Let’s compute the derivative:

### Function:
\[
y = 3 - \frac{5}{x^2} + \frac{2x^{1/2}}{5}
\]

### Step-by-step differentiation:

1. The derivative of \( 3 \) is \( 0 \) (Constant Rule).
2. The derivative of \( -\frac{5}{x^2} \) can be rewritten as \( -5x^{-2} \). Using the power rule, the derivative is:
   \[
   -5 \times (-2)x^{-3} = 10x^{-3}
   \]
3. The derivative of \( \frac{2x^{1/2}}{5} \) is \( \frac{2}{5}x^{1/2} \). Again, using the power rule:
   \[
   \frac{2}{5} \times \frac{1}{2}x^{-1/2} = \frac{1}{5}x^{-1/2}
   \]

### Final derivative:
\[
\frac{dy}{dx} = 10x^{-3} + \frac{1}{5}x^{-1/2}
\]

Would you like any further details or clarifications?

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Here are five related questions to expand on this:

1. How does the power rule apply when differentiating functions with negative exponents?
2. What is the process for finding higher-order derivatives of similar functions?
3. How can you differentiate a function involving a product of two functions?
4. How would the chain rule apply if this function involved a composition of functions?
5. How would the integral of the original function be computed?

**Tip**: Remember that simplifying expressions before differentiation often makes the process easier.

Learn how to find the derivative of the function y=3 - 5/x^2 + 2x^1/2/5 using the Power and Quotient Rules. This solution involves applying basic calculus techniques, including the power rule and the quotient rule, to differentiate complex functions. Suitable for high school students in Grades 11-12 studying Calculus.

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