Introduction

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

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

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

$$\frac{dy}{dx}$$

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

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

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.