The Notation of the Differential Calculus

Excerpts

• The Notation of the Differential Calculus

  When any quantity is increased by an increment, which, consistently with the conditions of the problem, may be supposed as small as we please, this increment is denoted, not by a separate letter, but by prefixing the letter $d$, either followed by a full stop or not, to that already used to signify the quantity.

• The Notation of the Differential Calculus

  Thus, if $x$ becomes $x + dx$, $x^{2}$ becomes $x^{2} + d.x^{2}$. But this is also $(x + dx)^{2}$ or $x^{2} + 2x\, dx + (dx)^{2}$; whence $d.x^{2} = 2x\, dx + (dx)^{2}$.

• The Notation of the Differential Calculus

  Care must be taken not to confound $d.x^{2}$, the increment of $x^{2}$, with $(dx)^{2}$, or, as it is often written, $dx^{2}$, the square of the increment of $x$.

• The Notation of the Differential Calculus

  It must not be imagined that because $x$ occurs in the symbol $dx$, the value of the latter in any way depends upon that of the former: both the first value of $x$, and the quantity by which it is made to differ from its first value, are at our pleasure, and the letter $d$ must merely be regarded as an abbreviation of the words “*difference of*.”

• The Notation of the Differential Calculus

  If $x = 4$ and $dx = .01$, then $dy = .0801$ and $\dfrac{dy}{dx} = 8.01$. If $dx = .0001$, $dy = .00080001$ and $\dfrac{dy}{dx} = 8.0001$.

• The Notation of the Differential Calculus

  We must adopt the first of these explanations when $dy$ and $dx$ appear in a fraction, and the second when they are on opposite sides of an equation.

• The Notation of the Differential Calculus

  *The equations which we thus use are not absolutely true in any case, but may be brought as near as we please to the truth*, by making $dy$ and $dx$ sufficiently small.

• The Notation of the Differential Calculus

  In the Differential Calculus, the limit of the ratio only is retained, to the exclusion of the rest, which may be explained in either of the two following ways:

Equations

• The Notation of the Differential Calculus

  d.x^{2} = 2x\, dx + (dx)^{2}

  The increment of x squared, found by expanding (x + dx)^2, equals 2x dx plus the square of dx.

• The Notation of the Differential Calculus

  \dfrac{d.x^{2}}{dx} = 2x + dx

  Dividing the increment of x^2 by the increment of x gives 2x plus dx, which tends to 2x as dx diminishes.

• The Notation of the Differential Calculus

  \dfrac{d.x^{2}}{dx} = 2x

  The ratio of the increment of x^2 to the increment of x is, in the limit, 2x; in the first explanation this equation is strictly true.

• The Notation of the Differential Calculus

  \dfrac{dy}{dx} = 2x + dx

  For y = x^2 the ratio of the increment of y to the increment of x equals 2x plus dx.

• The Notation of the Differential Calculus

  \dfrac{dy}{dx} = 2x

  The ratio of the increment of y to the increment of x, with dx made zero, is 2x; this is the limit of the ratio, not a ratio of two actual quantities.

• The Notation of the Differential Calculus

  dy = 2x\, dx + (dx)^{2}

  The increment of y for y = x^2 is 2x dx plus the square of dx; the (dx)^2 term affects dy but not the limit of dy/dx.

Problems

No exercises in this chapter.