Recapitulation of Results Reached in the Theory of Functions

Excerpts

• Recapitulation of Results Reached in the Theory of Functions

  The following is a recapitulation of the principal results which have hitherto been noticed in the general theory of functions:

• Recapitulation of Results Reached in the Theory of Functions

  That if in the equation $y = \phi(x)$, the variable $x$ receives an increment $dx$, $y$ is increased by the series ’ x  dx + ” x  (dx)^22 + ”’ x  (dx)^32·3 + etc.

• Recapitulation of Results Reached in the Theory of Functions

  $\phi' x$ is the limit of $\dfrac{dy}{dx}$, or the quantity to which the latter will approach, and to which it may be brought as near as we please, when $dx$ is diminished.

• Recapitulation of Results Reached in the Theory of Functions

  That $\phi'' x$ is derived in the same manner from $\phi' x$, that $\phi' x$ is from $\phi x$; viz., that in like manner as $\phi' x$ is the coefficient of $dx$ in the development of $\phi(x + dx)$, so $\phi'' x$ is the coefficient of $dx$ in the development of $\phi'(x + dx)$; similarly $\phi''' x$ is the coefficient of $dx$ in the development of $\phi''(x + dx)$, and so on.

• Recapitulation of Results Reached in the Theory of Functions

  That in every case which occurs in practice, $dx$ may be taken so small, that any term of the series above written may be made to contain the aggregate of those which follow, as often as we please; whence, though $\phi' x\, dx$ is not the actual increment produced by changing $x$ into $x + dx$ in the function $\phi x$, yet, by taking $dx$ sufficiently small, it may be brought as near as we please to a ratio of equality with the actual increment.

• Recapitulation of Results Reached in the Theory of Functions

  It is called the differential coefficient of $y$.

Equations

• This equation is in Rational Explanation of the Language of Leibnitz (Rational Explanation of the Language of Leibnitz)

• Recapitulation of Results Reached in the Theory of Functions

  \phi' x\, dx + \phi'' x\, \frac{(dx)^{2}}{2} + \phi''' x\, \frac{(dx)^{3}}{2·3} + \etc.

  When x receives an increment dx, the increase in y is given by a series in powers of dx whose coefficients are the successive derived functions of phi at x.

Problems

No exercises in this chapter.