One of the most concise presentations
of the derivative concept according to the classic differential
calculus may be found into The Engineer’s Textbook^{1}8.

**Derivative’s definition**.
Considering *y=f(x)* as a continuous function within an interval
*(a, b)* and a point *x*_{0} inside it. By
definition, it is named the function’s derivative into *x*_{0}
the limit towards which the ratio between the function growth and
variable growth tends to, when the latter tends to zero.

(X.3.2.2.1)

If this limit really exists, we may say
that the function *f(x)* is derivable in *x*_{0}.
If we are making the graphical plotting of the function *f(x)*,
the derivative represents in a specific point an angular coefficient
of the tangent to the curve. It is possible that the limit of this
ratio to have two values in a single point, just as
tends to zero by means of positive or negative values; we may say
that we are dealing with a derivative oriented either to the left or
to the right.

**Differentials. **We shall consider
*y=f(x)* as a derivable function into an interval *(a,b)*
by considering *x* a variable ranging within this interval. The
growth of variable *dx* shall be called the variable’s
differential. By definition, we shall assign the following value to
the function’s differential:

(X.3.2.2.2)

In order to comment the above-mentioned definitions, the following elements are being displayed in the figure X.3.2.2.1:

the curve

*f(x)*on which there is a current point ;there are other two points on the curve and, where and ;

tangent at the curve

*f(x)*in the point P, where the points and may be found on;

*Fig. X.3.2.2.1*

One may observe that the points *M, P*
and *N* belong to the curve *f(x)*, whereas the points Q
and S do not, but when making the differentials calculation on the
left or right side of the point *P* even the values from Q and S
are intervening, although they do not belong to the function (for
example, the differential on the right side of *P* is
represented by the segment *RQ*, according to the definition,
whereas the real variation is RN). In case that the values of the
function from *M, P* and *N* are obtained by means of
sampling, the values from *Q* and *S* are clearly out of
question since they are pure abstract values (generated by means of
calculus). In the plot from the figure X.3.2.2.1, the line which
connects two points from the curve, for instance *PM*, is
considered as the *secant* of the curve *f(x)*. This secant
shows an angular direction
against the reference axis *X* (axis of the values belonging to
the independent variable), evaluated on the left of the reference
,
given by the following relation:

(X.3.2.2.3)

and secant *PN* has an angular
direction
against the same reference axis *X*, evaluated on the right of
the reference
,
given by the following relation:

(X.3.2.2.4)

Both relations are valid for any __finite
and non-zero interval__
,
but according to the relation X.3.2.2.1, when defining the classic
derivative, this interval becomes null at a certain point (in the
point *P*), where the classic differential calculus defines the
__derivative in the point __* P*.

18 *** - Manualul Inginerului - Editura Tehnică, București, 1965.

Copyright © 2006-2011 Aurel Rusu. All rights reserved.