The given function is:

$y = -\frac{\sqrt{x^3}}{2}$

We need to find the derivative $\frac{dy}{dx}$ and express the answer in radical form, simplifying all fractions.

Step 1: Rewrite the expression

First, express $\sqrt{x^3}$ in terms of exponents. Recall that $\sqrt{x^3} = x^{\frac{3}{2}}$, so the function becomes:

$y = -\frac{1}{2} x^{\frac{3}{2}}$

Step 2: Apply the power rule for differentiation

The power rule for differentiation states that:

$\frac{d}{dx} x^n = n x^{n-1}$

Using this rule, differentiate $y = -\frac{1}{2} x^{\frac{3}{2}}$:

$\frac{dy}{dx} = -\frac{1}{2} \times \frac{3}{2} x^{\frac{3}{2} - 1}$

Step 3: Simplify the expression

Simplify the coefficient:

$-\frac{1}{2} \times \frac{3}{2} = -\frac{3}{4}$

Now, simplify the exponent:

$\frac{3}{2} - 1 = \frac{1}{2}$

Thus, the derivative is:

$\frac{dy}{dx} = -\frac{3}{4} x^{\frac{1}{2}}$

Step 4: Convert back to radical form

Recall that $x^{\frac{1}{2}}$ is the same as $\sqrt{x}$. Therefore, the final answer is:

$\frac{dy}{dx} = -\frac{3}{4} \sqrt{x}$

Final Answer:

$\boxed{-\frac{3}{4} \sqrt{x}}$

Would you like further details or have any questions?

Here are 5 related questions you might explore:

1. What is the general power rule for derivatives, and how is it applied?
2. How do you differentiate functions with fractional exponents?
3. What is the significance of simplifying expressions with radicals in calculus?
4. How would the result change if the function had a different coefficient?
5. Can you apply the chain rule if the function had an inner function instead of just $x^3$?

Tip: When working with exponents in calculus, rewriting radicals in terms of fractional powers can make differentiation and integration easier.