Question

If the rate of change in `x` is `"3 s"^(-1)`, and `(dy)/(dx) = 5`, what is the rate of change in `y`? Is `y` changing faster than `x` or vice versa?

Answer 1

The rate of change, or derivative, of `y` with respect to `x` can be written as

`(dy)/(dx) = 5`,

or

`dy = 5dx`.

We can say that on a non-infinitesimally-small scale,

`Deltay` `~~` `5Deltax`.

Therefore, if `x` changes as `"3 s"^(-1)`, then we let `Deltax = 3`, and

`Deltay = 5xx3 = "15 s"^(-1) > Deltax`,

and we have that `color(blue)(Deltay > Deltax)`.

This means `y` is changing faster than `x`, if we assume that the change in `x` does not itself change, and the change in `y` does not itself change either.