Thus far we have represented the derivative of $ y = f(x)$ by the notation

$\displaystyle \frac{dy}{dx} = f'(x).

We have taken special pains to impress on the student that the symbol

$\displaystyle \frac{dy}{dx}

was to be considered not as an ordinary fraction with $ dy$ as numerator and $ dx$ as denominator, but as a single symbol denoting the limit of the quotient

$\displaystyle \frac{\Delta y}{\Delta x}

as $ \Delta x$ approaches the limit zero.

Problems do occur, however, where it is very convenient to be able to give a meaning to $ dx$ and $ dy$ separately, and it is especially useful in applications of the Integral Calculus. How this may be done is explained in what follows.

david joyner 2008-11-22