Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/51

Rh It is apparent that as $$\scriptstyle{\Delta x}$$ decreases, $$\scriptstyle{\Delta y}$$ also diminishes, but their ratio takes on the successive values $$\scriptstyle{9,~8.8,~8.6,~8.4,~8.2,~8.1,~8.01}$$; illustrating the fact that $$\scriptstyle{\frac{\Delta y}{\Delta x}}$$ can be brought as near to $$\scriptstyle{8}$$ in value as we please by making $$\scriptstyle{\Delta x}$$ small enough. Therefore

The fundamental definition of the Differential Calculus is:

When the limit of this ratio exists, the function is said to be differentiable, or to possess a derivative.

The above definition may be given in a more compact form symbolically as follows: Given the function Rhand consider $$\scriptstyle{x}$$ to have a fixed value.

Let $$\scriptstyle{x}$$ take on an increment $$\scriptstyle{\Delta x}$$; then the function $$\scriptstyle{y}$$ takes on an increment $$\scriptstyle{\Delta y}$$, the new value of the function being Rh

To find the increment of the function, subtract (A) from (B), giving Rh

Dividing by the increment of the variable, $$\scriptstyle{\Delta x}$$, we get Rh

The limit of this ratio when $$\scriptstyle{\Delta x}$$ approaches the limit zero is, from our definition, the derivative and is denoted by the symbol $$\scriptstyle{\frac{dy}{dx}}$$. Therefore Rh defines the derivative of $$\scriptstyle{y}$$ [or $$\scriptstyle{f(x)}$$] with respect to $$\scriptstyle{x}$$.