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

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DIFFERENTIATION

27

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

$\scriptstyle {{\underset {\Delta x=0}{\operatorname {limit} }}\;{\frac {\Delta y}{\Delta x}}=8}$ .^[1]

28. Derivative of a function of one variable. The fundamental definition of the Differential Calculus is:

The derivative^[2] of a function is the limit of the ratio of the increment of the function to the increment of the independent variable, when the latter increment varies and approaches the limit zero.

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

(A)

\scriptstyle {y=f(x)}

,

and 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

(B)

\scriptstyle {y+\Delta y=f(x+\Delta x)}

.

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

(C)

\scriptstyle {\Delta y=f(x+\Delta x)-f(x)}

.

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

(D)

\scriptstyle {{\frac {\Delta y}{\Delta x}}={\frac {f(x+\Delta x)-f(x)}{\Delta x}}}

.

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

(E)

\scriptstyle {{\frac {dy}{dx}}={\underset {\Delta x=0}{\operatorname {limit} }}\;{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}

defines the derivative of $\scriptstyle {y}$ [or $\scriptstyle {f(x)}$ ] with respect to $\scriptstyle {x}$ .

↑ The student should guard against the common error of concluding that because the numerator and denominator of a fraction are each approaching zero as a limit, the limit of the value of the fraction (or ratio) is zero. The limit of the ratio may take on any numerical value. In the above example the limit is $\scriptstyle {8}$ .
↑ Also called the differential coefficient or the derived function.

[1] The student should guard against the common error of concluding that because the numerator and denominator of a fraction are each approaching zero as a limit, the limit of the value of the fraction (or ratio) is zero. The limit of the ratio may take on any numerical value. In the above example the limit is $\scriptstyle {8}$ .

[2] Also called the differential coefficient or the derived function.

[1]

[2]