Question

Find \(\frac {dy}{dx}\): \(x^2+xy+y^2=100\)

Answer 1

The given relationship is x² + xy + y² = 100
Differentiating this relationship with respect to x, we obtain
\(\frac {d}{dx}\)(x² + xy + y²) = \(\frac {d}{dx}\)(100)

⇒ \(\frac {d}{dx}\)(x²) + \(\frac {d}{dx}\)(xy) + \(\frac {d}{dx}\)(y²)=0

⇒ 2x + [y . \(\frac {d}{dx}\)(x) + x . \(\frac {dy}{dx}\)] + 2y \(\frac {dy}{dx}\) = 0         [using product rule and chain rule]

⇒ 2x + y . 1 + x . \(\frac {dy}{dx}\) + 2y \(\frac {dy}{dx}\) = 0

⇒ 2x + y + (x+2y) \(\frac {dy}{dx}\) = 0

∴ \(\frac {dy}{dx}\) = \(-\frac {2x+y}{x+2y}\)