If \(x^3 + y^3 = 9\), then the value of \(\displaystyle\frac{dy}{dx}\) at the point (1, 2) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(-\displaystyle\frac{1}{4}\)
Since y is a function of x, the term \(y^3\) is differentiated by the chain rule (implicit differentiation):
\( \displaystyle 3x^2 + 3y^2\frac{dy}{dx} = 0 \)
\( \displaystyle \frac{dy}{dx} = -\frac{x^2}{y^2} \)
The point (1, 2) lies on the curve, since 1 + 8 = 9. At this point:
\( \displaystyle \frac{dy}{dx} = -\frac{1}{4} \)
Why the other options are incorrect
Option (1) \(-\displaystyle\frac{1}{2}\) — Incorrect. This is \(-\frac{x}{y}\); the squares have been lost.
Option (2) \(\displaystyle\frac{1}{4}\) — Incorrect. The minus sign is missing; y must decrease as x increases on this curve.
Option (4) −4 — Incorrect. This is \(-\frac{y^2}{x^2}\), which is \(\frac{dx}{dy}\), not \(\frac{dy}{dx}\).