Question

If \[ 2x^2 - 3xy + 4y^2 + 2x - 3y + 4 = 0 \] then \[ \left( \frac{dy}{dx} \right)_{(3,2)} = \]

• A. \( -5 \)
• B. \( \frac{5}{7} \)
• C. \( -2 \)
• D. \( \frac{2}{7} \)

Hint:

Implicit differentiation is useful when \( y \) is not explicitly solved in terms of \( x \).

Answer 1

The Correct Option is C

Step 1: Implicit Differentiation.
Differentiating both sides with respect to \( x \): \[ \frac{d}{dx} \left( 2x^2 - 3xy + 4y^2 + 2x - 3y + 4 \right) = 0 \] Using the product rule for \( -3xy \): \[ 4x - 3(y + x \frac{dy}{dx}) + 8y \frac{dy}{dx} + 2 - 3 \frac{dy}{dx} = 0 \] Step 2: Solve for \( \frac{dy}{dx} \).
\[ (4x + 2) - 3y - 3x \frac{dy}{dx} + 8y \frac{dy}{dx} - 3 \frac{dy}{dx} = 0 \] \[ 4x + 2 - 3y = (3x - 8y + 3) \frac{dy}{dx} \] Substituting \( (x, y) = (3,2) \): \[ (4(3) + 2 - 3(2)) = (3(3) - 8(2) + 3) \frac{dy}{dx} \] \[ (12 + 2 - 6) = (9 - 16 + 3) \frac{dy}{dx} \] \[ 8 = (-4) \frac{dy}{dx} \] \[ \frac{dy}{dx} = -2 \] Thus, the required derivative is: \[ \mathbf{-2} \]