**Answer:**

To find the derivative of Y with respect to x, we need to apply the chain rule. Let's break down the given expression and then differentiate it step by step.

The expression Y is given as:

Y = [(x^2 a^2)^1/2 / (x^2 b^2)^1/2]^-1

**Step 1: Simplify the expression**

To simplify the expression, we can rewrite it as:

Y = [(x^2 a^2) / (x^2 b^2)]^-1/2

**Step 2: Apply the power rule**

To differentiate Y, we need to apply the power rule, which states that the derivative of x^n with respect to x is n * x^(n-1).

Let's differentiate the expression with respect to x:

dY/dx = d/dx [(x^2 a^2) / (x^2 b^2)]^-1/2

**Step 3: Apply the chain rule**

To apply the chain rule, we need to differentiate the outer function first and then multiply it by the derivative of the inner function.

Let's differentiate the outer function [(x^2 a^2) / (x^2 b^2)]^-1/2:

d/dx [(x^2 a^2) / (x^2 b^2)]^-1/2

= -1/2 * [(x^2 a^2) / (x^2 b^2)]^-3/2 * d/dx [(x^2 a^2) / (x^2 b^2)]

**Step 4: Differentiate the inner function**

To differentiate the inner function, we need to apply the quotient rule, which states that the derivative of (f/g) with respect to x is (g * df/dx - f * dg/dx) / g^2.

Let's differentiate [(x^2 a^2) / (x^2 b^2)] with respect to x:

d/dx [(x^2 a^2) / (x^2 b^2)]

= [(x^2 b^2) * 2x * a^2 - (x^2 a^2) * 2x * b^2] / (x^2 b^2)^2

= 2x * [(x^2 b^2 * a^2 - x^2 a^2 * b^2) / (x^2 b^2)^2]

= 2x * [(b^2 a^2 - a^2 b^2) / (b^4 x^2)]

= 2x * (a^2 - b^2) / (b^4 x^2)

**Step 5: Substitute the inner derivative back into the chain rule expression**

Now, we can substitute the derivative of the inner function back into the chain rule expression:

dY/dx = -1/2 * [(x^2 a^2) / (x^2 b^2)]^-3/2 * 2x * (a^2 - b^2) / (b^4 x^2)

= -x * (a^2 - b^2) / [(x^2 a^2) / (x^2 b^2)]^(3