## Abstract

**Derivative of a function of one variable**(

^{1}). The

*derivative*of a function

*y*=

*f*(

*x*) (denoted by \({y}',\;y^{\circ},\;Dy,\;\frac{dy}{dx},\;{f}'(x),\;Df(x),\;\frac{df(x)}{dx}\)) is equal, for a given value of

*x*, to the limit of the ratio of the increment

*Δy*of the function to the increment

*Δx*of the argument, when

*Δx*tends to zero:

$${f}'(x)=\lim_{\mathit{\Delta} x \to0}\frac{f(x+\mathit{\Delta} x)-f(x)}{\mathit{\Delta} x}$$

## Keywords

Partial Derivative Tangent Line Differential Calculus Total Differential Geometric Significance## Preview

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## Notes

- (1).Only single-valued functions will be considered in this chapter.Google Scholar
- (2).The formula
*f*′(*x*) holds only in the case when a common unit of measure is chosen on the*x*and*y*axes.Google Scholar - (1).For the total differential in vector form see p. 633, theory of field.Google Scholar
- (2).See p. 363.Google Scholar
- (3).This notation is convenient only in the case when
*x*is an independent variable. It is inconvenient, for example, when*x*= φ(υ); see p. 373, change of variables.Google Scholar - (1).This notation is convenient only in the case, when
*x*is an independent variable. It is inconvenient when*x*= φ(υ) (see p. 373, change of variables).Google Scholar - (2).When
*x*and*y*are functions of new variables, the formulas are more complicated; see pp. 375, 376.Google Scholar - (1).If the formula of transformation is given in implicit form Φ(
*x*,*t*) = 0, then the derivatives \(\frac{dy}{dx},\;\frac{d^{2}y}{dx^{2}},\;\frac{d^{3}y}{dx^{3}}\) are computed by the same formula, but φ′(*t*), φ″(*t*), φ″′(*t*) are computed as derivatives of an implicit function; if the expression involves the variable*x*, then*x*should be eliminated by means of the equation Φ(*x*,*t*) = 0.Google Scholar - (1).See p. 645.Google Scholar
- (1).I.e., at the points which are not end points of the interval.Google Scholar
- (2).This condition is valid for functions monotonely increasing in the wider sense (see p. 326). In order that the function increase or decrease strictly monotonely, another condition should be added: that the derivative
*f*′(*x*) does not identically vanish in any interval contained in the domain of existence. This condition is not fulfilled, for example, on the segment*BC*in Fig. 290b.Google Scholar - (1).In the case of a strictly monotone function, the tangent can be parallel to the
*x*axis only at isolated points (e.g., the point*A*, Fig. 290a), but not in a whole interval (*BC*in Fig. 290b).Google Scholar - (2).I.e., a point which is not an end-point of the interval.Google Scholar
- (1).
*h*can be here positive as well as negative.Google Scholar - (2).In mathematical analysis maxima and minima have a common name of “turning values” or “extrema”.Google Scholar
- (1).This concept of the remainder of a Taylor series does not coincide with that of the remainder of a series as introduced on p. 355. Both concepts are the same only in cases for which the formula (T) is true.Google Scholar

## Copyright information

© Verlag Harri Deutsch, Zürich 1973