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The given function is:

\[
y = \frac{510}{\sqrt{x}}
\]

To find \( dy \) (or \( \frac{dy}{dx} \)), we need to take the derivative of the function with respect to \( x \).

We can rewrite \( \frac{510}{\sqrt{x}} \) as \( 510x^{-\frac{1}{2}} \). Now, differentiate using the power rule, which states:

\[
\frac{d}{dx}(x^n) = n x^{n-1}
\]

Applying the power rule:

\[
\frac{dy}{dx} = 510 \cdot \left( -\frac{1}{2} \right) x^{-\frac{3}{2}} = -255 x^{-\frac{3}{2}}
\]

Thus, the simplified derivative is:

\[
dy = \frac{-255}{x^{\frac{3}{2}}}
\]

Would you like any further details or clarifications?

Here are 5 related questions to expand your understanding:
1. What is the significance of rewriting the square root in terms of exponents for differentiation?
2. How would you apply the chain rule if the function were more complex?
3. What does the negative exponent in the derivative signify in terms of the function's behavior?
4. How would the derivative change if the numerator was a function of \( x \) rather than a constant?
5. How can this result be interpreted in the context of related rates or physical applications?

**Tip**: When differentiating powers of \( x \), always check if the exponent can be rewritten to simplify the use of the power rule.

Learn how to differentiate the function y = 510 / √x using the power rule. This problem explores the concepts of differentiation, exponents, and the application of the power rule. Suitable for students in Grades 10-12.

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