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: $\boxed{3}$.