Differentials

It is so small, it is always smaller than what you think it might be. By itself, a differential is too small to be a recognizable nunbet, but when paired up with another differential to make a ratio, you might get something fairly easy to recognize:

y = 3x

\dfrac {dy}{dx} = 3

Ratios of Differentials

The ratio of differentials can be a constant or a function.

Question: What is y if \dfrac {dy}{dx} = x^2?

First we rearrange the equation to have differentials on both sides:

dy = x^2 dx

\int dy = \int x^2 dx

y + c_1= \dfrac {1}{3} x^3 + c_2

let c_3 = c_2 - c_1

y = \dfrac {1}{3} x^3 + c_3