Answer is: option3
\( \frac{5\sqrt{2}}{2} \)Solution:
Given equation: \[ 3x^4 - x^2 - y^2 = 0 \] Differentiating both sides implicitly: \[ 12x^3 - 2x - 2y \frac{dy}{dx} = 0 \] \[ -2y \frac{dy}{dx} = -12x^3 + 2x \] \[ \frac{dy}{dx} = \frac{12x^3 - 2x}{2y} \] \[ = \frac{6x^3 - x}{y} \] Substituting \( x = 1, y = \sqrt{2} \): \[ \frac{dy}{dx} = \frac{6(1)^3 - 1}{\sqrt{2}} \] \[ = \frac{6 - 1}{\sqrt{2}} \] \[ = \frac{5}{\sqrt{2}} \] Rationalizing the denominator: \[ \frac{5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \] Final Answer: \[ \boxed{\frac{5\sqrt{2}}{2}} \]