Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/53

The symbol $D_{x}$ is used by some writers instead of ${\frac {d}{dx}}$. If then

$y=f(x),$

we may write the identities

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

30. Differentiable functions. From the Theory of Limits it is clear that if the derivative of a function exists for a certain value of the independent variable, the function itself must be continuous for that value of the variable.

The converse, however, is not always true, functions having been discovered that are continuous and yet possess no derivative. But such functions do not occur often in applied mathematics, and in this book only differentiable functions are considered, that is, functions that possess a derivative for all values of the independent variable save at most for isolated values.

31. General rule for differentiation. From the definition of a derivative it is seen that the process of differentiating a function $y=f(x)$ consists in taking the following distinct steps:

General Rule for Differentiation^[1]

First Step. In the function replace $x$ by $x+\Delta x$, giving a new value of the function, $y+\Delta y$.
Second Step. Subtract the given value of the function from the new value in order to find $\Delta y$ (the increment of the function).
Third Step. Divide the remainder $\Delta y$ (the increment of the function) by $\Delta x$ (the increment of the independent variable).
Fourth Step. Find the limit of this quotient, when $\Delta x$ (the increment of the independent variable) varies and approaches the limit zero. This is the derivative required.

The student should become thoroughly familiar with this rule by applying the process to a large number of examples. Three such examples will now be worked out in detail.

Illustrative Example 1. Differentiate $3x^{2}+5$.

Solution. Applying the successive steps in the General Rule, we get, after placing

	$y$	$=3x^{2}+5$,
First step.	$y+\Delta y$	$=3(x+\Delta x)^{2}+5$
		$=3x^{2}+6x\cdot \Delta x+3(\Delta x)^{2}+5$.

1. Also called the Four-step Rule.