Taylor's Theorem. Derived Functions

Excerpts

• Taylor's Theorem. Derived Functions

  Thus, when $\phi x = x^{n}$, $\phi' x = nx^{n-1}$, $\phi'' x = n(n - 1)x^{n-2}$, etc.; when $\phi x = \sin x$, $\phi' x = \cos x$, $\phi'' x = -\sin x$, etc.

• Taylor's Theorem. Derived Functions

  Here the coefficient of $h$ is $-\dfrac{1}{x^{2}}$, which is the same as $\phi'' x$ in the third example.

• Taylor's Theorem. Derived Functions

  The proof of this is equivalent to *Taylor’s Theorem* already alluded to (15); and the fact may be verified in the examples already given.

• Taylor's Theorem. Derived Functions

  This never happens in the developments which we shall be required to consider in the Differential Calculus.

• Taylor's Theorem. Derived Functions

  It appears, then, that the development of $\phi(x + h)$ consists of certain functions of $x$, the first of which is $\phi x$ itself, and the remainder of which are multiplied by $h$, $\dfrac{h^{2}}{2}$, $\dfrac{h^{3}}{2·3}$, $\dfrac{h^{4}}{2·3·4}$, and so on.

• Taylor's Theorem. Derived Functions

  The following relation exists between $\phi x$, $\phi' x$, $\phi'' x$, etc. In the same manner as $\phi' x$ is the coefficient of $h$ in the development of $\phi(x + h)$, so $\phi'' x$ is the coefficient of $h$ in the development of $\phi'(x + h)$, and $\phi''' x$ is the coefficient of $h$ in the development of $\phi''(x + h)$; $\phi^{\text{iv}} x$ is the coefficient of $h$ in the development of $\phi'''(x + h)$, and so on.

• Taylor's Theorem. Derived Functions

  In this case, by theorem (1*b*), the series is convergent; it follows, therefore, that a value of $h$ can always be found so small that $ph + qh^{2} + rh^{3} + \etc.$, shall be convergent, at least unless the coefficients $p$, $q$, $r$, etc., be such that the ratio of any one to the preceding increases without limit, as we take more distant terms of the series. This never happens in the developments which we shall be required to consider in the Differential Calculus.

• Taylor's Theorem. Derived Functions

  When $\phi x = a^{x}$, $\phi' x = ka^{x}$, and $\phi'(x + h) = ka^{x+h} = k(a^{x} + ka^{x}\, h + \etc.)$. The coefficient of $h$ is here $k^{2} a^{x}$, which is the same as $\phi'' x$.

• Taylor's Theorem. Derived Functions

  Again, if $\phi x = \log x$, $\phi' x = \dfrac{1}{x}$, and $\phi'(x + h) = \dfrac{1}{x + h} = \dfrac{1}{x} - \dfrac{h}{x^{2}} + \etc.$, as appears by common division. Here the coefficient of $h$ is $-\dfrac{1}{x^{2}}$, which is the same as $\phi'' x$ in the third example.

Equations

• Taylor's Theorem. Derived Functions

  (x + h)^{n} = x^{n} + nx^{n-1}h + n(n - 1)x^{n-2} \frac{h^{2}}{2} + n(n - 1)(n - 2)x^{n-3} \frac{h^{3}}{2·3}

  The power (x + h)^n is developed in ascending powers of the increment h, with the coefficients of h, h^2/2 and h^3/(2·3) given.

• Taylor's Theorem. Derived Functions

  \log(x + h) = \log x + \frac{1}{x}\, h - \frac{1}{x^{2}}\, \frac{h^{2}}{2} + \frac{2}{x^{3}}\, \frac{h^{3}}{2·3}

  The logarithm of x + h is developed in powers of the increment h.

• Taylor's Theorem. Derived Functions

  \cos(x + h) = \cos x - \sin x\, h - \cos x\, \frac{h^{2}}{2} + \sin x\, \frac{h^{3}}{2·3}

  The cosine of x + h is developed in powers of h; the terms are positive and negative in pairs.

• Taylor's Theorem. Derived Functions

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

  A function phi(x + h) is expanded as a series in powers of h whose coefficients are the successive derived functions of phi at x, divided by factorials.

• Taylor's Theorem. Derived Functions

  \phi' x = nx^{n-1}

  When phi x = x^n, its first derived function is n x^(n-1).

• Taylor's Theorem. Derived Functions

  \phi'' x = n(n - 1)x^{n-2}

  When phi x = x^n, its second derived function is n(n - 1) x^(n-2).

• Taylor's Theorem. Derived Functions

  \phi' x = \cos x

  When phi x = sin x, its first derived function is cos x.

• Taylor's Theorem. Derived Functions

  \phi'' x = -\sin x

  When phi x = sin x, its second derived function is minus sin x.

• Taylor's Theorem. Derived Functions

  \phi' x = ka^{x}

  When phi x = a^x, its first derived function is k a^x, where k is the natural logarithm of a.

• Taylor's Theorem. Derived Functions

  \phi'(x + h) = ka^{x+h} = k(a^{x} + ka^{x}\, h + \etc.)

  The first derived function of a^x, evaluated at x + h, is k a^(x+h), developed in powers of h.

• Taylor's Theorem. Derived Functions

  \dfrac{1}{x + h} = \dfrac{1}{x} - \dfrac{h}{x^{2}} + \etc.

  The reciprocal 1/(x + h) is developed in powers of h, found by common division.

• Taylor's Theorem. Derived Functions

  \phi''(x + h) = -\dfrac{1}{(x + h)^{2}} = -(x + h)^{-2}

  The second derived function of log x, evaluated at x + h, equals minus one over (x + h) squared.

Problems

No exercises in this chapter.