may be made smaller than any given positive number by making sufficiently small. If we generally denote by an expression having such a property, we can write
This allows us to express the increment of the function in the form
This result may be uttered as the
Theorem. If the derivative exists, then the increment of the function corresponding to the increment of the argument from to may be divided into two essentially different parts:
. One part is proportional to the increment of the argument, i.e. it equals this increment multiplied by a coefficient which is on the increment.
. The ratio of the other part to the increment of the argument tends to 0 along with .
As well, the converse of the theorem is true.
By Leibniz, the former part is called the differential, or the differential increment of the function, and denoted by , briefly .
It is easily checked that when one has set the tangent line of the curve at the point , the differential increment geometrically means the increment of the ordinate of the corresponding to the transition from the abscissa to the ascissa .
The differential of the identity function (, ) is
Accordingly, one can without discrepancies denote the increment of the variable by . Therefore, the differential of a function gets the notation
Remark. One can write certain for forming differentials. For example, if and are two differentiable functions, one has
Naturally, they seem trivial consequences of the sum rule and the product rule, but they include a deeper contents in the case where and depend on more than one variable (see total differential (http://planetmath.org/TotalDifferential)).
As for a composite function, e.g. , the chain rule and (2) yield
- 1 Ernst Lindelöf: Johdatus korkeampaan analyysiin. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
- 2 E. Lindelöf: Einführung in die höhere Analysis. Nach der ersten schwedischen und zweiten finnischen Auflage auf deutsch herausgegeben von E. Ullrich. Teubner, Leipzig (1934).
|Date of creation||2014-02-23 15:34:46|
|Last modified on||2014-02-23 15:34:46|
|Last modified by||pahio (2872)|