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(A) The velocity $v$ of a particle is defined as the rate of change of displacement with respect to time,given by $v = \frac{dx}{dt}$.Given the displa

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$k\,{e^{ - bt}}$

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(A) The velocity $v$ of a particle is defined as the rate of change of displacement with respect to time,given by $v = \frac{dx}{dt}$.Given the displacement relation: $x = \frac{k}{b}\,[1 - {e^{ - bt}}]$.Differentiating $x$ with respect to $t$:$v = \frac{d}{dt} \left[ \frac{k}{b} (1 - e^{-bt}) \right]$$v = \frac{k}{b} \left[ \frac{d}{dt}(1) - \frac{d}{dt}(e^{-bt}) \right]$Since the derivative of a constant is $0$ and $\frac{d}{dt}(e^{-bt}) = -b e^{-bt}$:$v = \frac{k}{b} [0 - (-b) e^{-bt}]$$v = \frac{k}{b} [b e^{-bt}]$$v = k e^{-bt}$.Thus,the correct option is $A$.

The displacement of the particle varies with time according to the relation $x = \frac{k}{b}\,[1 - {e^{ - bt}}]$. Then the velocity of the particle is

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