Question

If \( y = x - x^2 \), then the rate of change of \( y^2 \) with respect to \( x^2 \) at \( x = 2 \) is: 

• A. \( 0 \)
• B. \( -1 \)
• C. \( 3 \)
• D. \( 9 \)

Hint:

To find the rate of change of one function with respect to another, use the chain rule and express derivatives in terms of the desired variable.

Answer 1

The Correct Option is C

Step 1: Expressing \( y^2 \)
Given \( y = x - x^2 \), we square both sides: \[ y^2 = (x - x^2)^2. \] Step 2: Differentiating Both Sides
Differentiating both sides with respect to \( x \): \[ \frac{d}{dx} (y^2) = 2y \cdot \frac{dy}{dx}. \] Now, differentiating \( y = x - x^2 \): \[ \frac{dy}{dx} = 1 - 2x. \] Step 3: Finding the Rate of Change of \( y^2 \) with Respect to \( x^2 \)
We need to find \( \frac{d(y^2)}{dx^2} \), which is: \[ \frac{d(y^2)}{dx^2} = \frac{\frac{d}{dx} (y^2)}{\frac{d}{dx} (x^2)}. \] Since \( \frac{d}{dx} (x^2) = 2x \), we substitute: \[ \frac{d(y^2)}{dx^2} = \frac{2y (1 - 2x)}{2x}. \] Step 4: Evaluating at \( x = 2 \)
First, find \( y \) at \( x = 2 \): \[ y = 2 - 2^2 = 2 - 4 = -2. \] Substituting \( x = 2 \) and \( y = -2 \): \[ \frac{d(y^2)}{dx^2} = \frac{2(-2) (1 - 4)}{2(2)} = \frac{2(-2)(-3)}{4} = \frac{12}{4} = 3. \] Final Answer: \( \boxed{3} \).

Answer 2

Step 1: Let’s define the functions 

• Given: \( y = x - x^2 \)
• We want: \( \frac{d(y^2)}{d(x^2)} \)

Step 2: Differentiate using the chain rule

Using the chain rule: \[ \frac{d(y^2)}{d(x^2)} = \frac{d(y^2)}{dx} \cdot \frac{dx}{d(x^2)} \] First: \[ \frac{d(y^2)}{dx} = 2y \cdot \frac{dy}{dx} \] and: \[ \frac{dx}{d(x^2)} = \frac{1}{2x} \] So: \[ \frac{d(y^2)}{d(x^2)} = \frac{2y \cdot \frac{dy}{dx}}{2x} = \frac{y \cdot \frac{dy}{dx}}{x} \]

Step 3: Evaluate at \( x = 2 \)

Given: \[ y = x - x^2 = 2 - 4 = -2 \] \[ \frac{dy}{dx} = \frac{d}{dx}(x - x^2) = 1 - 2x = 1 - 4 = -3 \] Now plug in: \[ \frac{d(y^2)}{d(x^2)} = \frac{(-2) \cdot (-3)}{2} = \frac{6}{2} = 3 \]

Answer:

\( \boxed{3} \)