1. **State the problem:** We are given the function $y = \frac{1}{\sqrt{x}}$ and need to express it in another form and find its derivative. 2. **Rewrite the function:** Recall that $\sqrt{x} = x^{\frac{1}{2}}$, so $$y = \frac{1}{\sqrt{x}} = \frac{1}{x^{\frac{1}{2}}} = x^{-\frac{1}{2}}.$$ 3. **Formula for derivative:** The power rule states that if $y = x^n$, then $$\frac{dy}{dx} = n x^{n-1}.$$ 4. **Apply the power rule:** Here, $n = -\frac{1}{2}$, so $$\frac{dy}{dx} = -\frac{1}{2} x^{-\frac{1}{2} - 1} = -\frac{1}{2} x^{-\frac{3}{2}}.$$ 5. **Rewrite the derivative:** Using radicals, $$\frac{dy}{dx} = -\frac{1}{2} \frac{1}{x^{\frac{3}{2}}} = -\frac{1}{2 x^{\frac{3}{2}}} = -\frac{1}{2 x \sqrt{x}}.$$ **Final answers:** - Equivalent form: $y = x^{-\frac{1}{2}}$ - Derivative: $\frac{dy}{dx} = -\frac{1}{2 x \sqrt{x}}$