Let us calculate $\frac{dy}{dx}$ for both parts.

a) $y = \frac{7}{x}$

Rewrite $y$ as $y = 7x^{-1}$. Now, use the power rule for differentiation:

$\frac{dy}{dx} = \frac{d}{dx}(7x^{-1}) = 7 \cdot (-1)x^{-2} = -\frac{7}{x^2}.$

b) $y = \frac{1}{4x^2}$

Rewrite $y$ as $y = \frac{1}{4}x^{-2}$. Again, apply the power rule for differentiation:

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

Final Results:

1. For $y = \frac{7}{x}$, $\frac{dy}{dx} = -\frac{7}{x^2}$.
2. For $y = \frac{1}{4x^2}$, $\frac{dy}{dx} = -\frac{1}{2x^3}$.

Let me know if you'd like further clarification or have additional questions!

Here are 5 related questions to extend your learning:

1. How does the chain rule apply when differentiating composite functions involving powers of $x$?
2. What is the interpretation of $\frac{dy}{dx}$ in real-world problems?
3. Can you explore higher-order derivatives of these functions?
4. How do you differentiate products or quotients of functions (product and quotient rules)?
5. What happens when you differentiate logarithmic or exponential functions?

Tip: