Differential Calculus

  • I. N. Bronshtein
  • K. A. Semendyayev


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


Partial Derivative Tangent Line Differential Calculus Total Differential Geometric Significance 
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  1. (1).
    Only single-valued functions will be considered in this chapter.Google Scholar
  2. (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
  3. (1).
    For the total differential in vector form see p. 633, theory of field.Google Scholar
  4. (2).
    See p. 363.Google Scholar
  5. (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
  6. (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
  7. (2).
    When x and y are functions of new variables, the formulas are more complicated; see pp. 375, 376.Google Scholar
  8. (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
  9. (1).
    See p. 645.Google Scholar
  10. (1).
    I.e., at the points which are not end points of the interval.Google Scholar
  11. (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
  12. (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
  13. (2).
    I.e., a point which is not an end-point of the interval.Google Scholar
  14. (1).
    h can be here positive as well as negative.Google Scholar
  15. (2).
    In mathematical analysis maxima and minima have a common name of “turning values” or “extrema”.Google Scholar
  16. (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

Authors and Affiliations

  • I. N. Bronshtein
  • K. A. Semendyayev

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