A particle moves along the curve \(y = x^{3} + 2x\). Its x-coordinate increases at the rate of 3 units per second. The rate at which its y-coordinate is changing when x = 1 is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 15 units per second
Both x and y depend on the time t. Differentiating \(y = x^{3} + 2x\) with respect to t by the chain rule:
\( \displaystyle \frac{dy}{dt} = (3x^{2} + 2)\frac{dx}{dt} \)
With x = 1 and \(\frac{dx}{dt} = 3\):
\( \displaystyle \frac{dy}{dt} = (3 + 2) \times 3 \)
\( = 15 \) units per second
Why the other options are incorrect
Option (1) 9 units per second — Incorrect. This is \(3x^{2} \times 3\): the derivative 2 of the term 2x is left out.
Option (2) 5 units per second — Incorrect. This is \(\frac{dy}{dx}\) at x = 1; it still has to be multiplied by \(\frac{dx}{dt} = 3\).
Option (3) 3 units per second — Incorrect. This is the rate of the x-coordinate, not of the y-coordinate.