## Quick Reference

Suppose that the quantity *y* is a function of the quantity *x*, so that *y*=*f*(*x*). If *f* is differentiable, the rate of change of *y* with respect to *x* is the derivative *dy*/*dx* or *f′*(*x*). The rate of change is often with respect to time *t*. Suppose, now, that *x* denotes the displacement of a particle, at time *t*, on a directed line with origin *O*. Then the velocity is *dx*/*dt* or ẋ, the rate of change of *x* with respect to *t*, and the acceleration is *d*^{2}*x*/*dt*^{2} or *ẍ*, the rate of change of the velocity with respect to *t*.

In the reference to velocity and acceleration in the preceding paragraph, a common convention has been followed, in which the unit vector **i** in the positive direction along the line has been suppressed. Velocity and acceleration are in fact vector quantities, and in the 1-dimensional case above are equal to ẋ**i** and *ẍ***i**. When the motion is in 2 or 3 dimensions, vectors are used explicitly. If **r** is the position vector of a particle, the velocity of the particle is the vector **r̈**̇ and the acceleration is **r̈**.

*Subjects:*
Mathematics.