\displaystyle 1.\ \text{If } y=f(x),\ \text{then }\frac{dy}{dx}\ \text{measures the rate of change of } y \text{ with respect to } x.

\displaystyle 2.\ \left(\frac{dy}{dx}\right)_{x=x_0}\ \text{represents the rate of change of } y \text{ with respect to } x \text{ at } x=x_0.

\displaystyle 3.\ \text{If the displacement of a particle moving in a straight line at time } t \text{ is given by } \\  s=f(t),\ \text{then}

\displaystyle (i)\ v=\text{Velocity at time } t=\frac{ds}{dt},\ a=\text{Acceleration at time } \\ t=\frac{dv}{dt}=\frac{d^2s}{dt^2}=v\frac{dv}{ds}

\displaystyle (ii)\ \text{If a particle moving in a straight line comes to rest, then } \\ \frac{ds}{dt}=0\ \text{and }\frac{d^2s}{dt^2}=0.

\displaystyle (iii)\ \text{If a particle moving in a straight line is instantaneously at rest, then } \\ \frac{ds}{dt}=0\ \text{but }\frac{d^2s}{dt^2}\neq0.

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