In Which We Give the Rules of This Calculus

Abstract

In Chapter 1 of the Analyse, l’Hôpital gives the rules of the differential calculus. His calculus is a calculus of equations and differentials, not of functions and derivatives. Modern readers will nevertheless recognize many familiar rules, including the product rule, the quotient rule, and the power rule. L’Hôpital has no need for the chain rule, because substitutions are handled in a very natural way in this calculus. Because readers in the 1690s may not have been familiar with negative and fractional exponents, l’Hôpital also explains those ideas here.

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Analysis

of the

Infinitely Small

First Part

On the Differential Calculus

FormalPara Definition I.

Those quantities are called variable which increase or decrease continually, as opposed to constant quantities that remain the same while others change. Thus in the parabola the ordinate and the abscissa are variable quantities, as opposed to the parameter, which is a constant quantity.¹

FormalPara Definition II.

[2] The infinitely small portion by which a variable quantity continually increases or decreases is called the Differential.² For example, let AMB be an arbitrary curved line (see Fig. 1.1) which has the line AC as its axis or diameter,³ and has PM as one of its ordinates. Let pm be another ordinate, infinitely close to the first one. Given this, if we also draw MR parallel to AC, and the chords AM Am, and describe the little circular arc MS of the circle with center A and radius AM, then Pp is the differential of AP, Rm the differential of PM, Sm the differential of AM, and Mm the differential of the arc AM. Furthermore, the little triangle MAm, which has the arc Mm as its base is the differential of the segment AM, and the little region MPpm is the differential of the region contained by the straight lines AP and PM, and by the arc AM.

Fig. 1.1

Definition of the Differential

FormalPara Corollary.

( §1 ) It is evident that the differential of a constant quantity is null or zero, or (what amounts to the same thing) that constant quantities do not have a differential.

FormalPara Note.

In what follows, we will make use of the symbol d to denote the differential of a variable quantity that is expressed by a single letter and, in order to avoid confusion, the letter d will not be used in any other way in the following calculations. If, for example, we denote AP by x, PM by y, AM by z, the arc AM by u, the curvilinear region APM by s, and the segment AM by t, then dx denotes the value of Pp, dy that of Rm, dz that of Sm, du that of the little arc Mm, ds that of the little region MPpm, and dt that of the little curvilinear triangle MAm.⁴

FormalPara Postulate I.

(§2). We suppose that two quantities that differ by an infinitely small quantity may be used interchangeably, or (what amounts to the same [3] thing) that a quantity which is increased or decreased by another quantity that is infinitely smaller than it is, may be considered as remaining the same. We suppose, for example, that we may take Ap for AP, pm for PM, the region Apm for the region APM, the little region MPpm for the little rectangle MPpR, the little sector AMm for the little triangle AMS, the angle pAm for the angle PAM, and so forth.

FormalPara Postulate II.

(§3) We suppose that a curved line may be considered as an assemblage of infinitely many straight lines, each one being infinitely small, or (what amounts to the same thing) as a polygon with an infinite number of sides, each being infinitely small, which determine the curvature of the line by the angles formed amongst themselves. We suppose, for example, that the portion Mm of the curve and the arc MS of the circle may be considered to be straight lines on account of their infinite smallness, so that the little triangle mSM may be considered to be rectilinear.

FormalPara Note.

In what follows, we will normally suppose that the final letters of the alphabet, z, y, x, etc., denote variable quantities, and conversely, that the first letters a, b, c, etc., denote constant quantities, so that as x becomes x+dx, y, z, etc., become y+dy, z+dz, etc., and a, b, c, etc. remain as a, b, c, etc. (see §1).

FormalPara Proposition I.

Problem. (§4) To take the differential of several quantities added together or subtracted from one another. ⁵

Let a + x + y − z be the expression whose differential we are to take. If we suppose that x is increased by an infinitely small quantity, i.e. that it becomes x +dx, then y [4] becomes y +dy, and z becomes z +dz. As for the constant a, it remains a (see §1). Thus, the given quantity a + x + y − z becomes a + x +dx + y +dy − z −dz, and the differential, which is found by subtracting the former from the latter, is dx +dy −dz. It is similar for other expressions, giving rise to the following rule.

FormalPara Rule I.

For quantities added or subtracted.

We take the differential of each term in the given quantity and, keeping the signs the same, we add them together in a new quantity which is the differential that we wish to find.

FormalPara Proposition II.

 Problem. (§5) To take the differential of a product composed of several quantities multiplied together.

• 1. The differential of xy is y dx + x dy. This is because y becomes y +dy while x becomes x +dx, and consequently xy becomes xy + y dx + x dy +dx dy, which is the product of x +dx and y +dy. The differential is y dx + x dy +dx dy, that is to say (see §2) y dx + x dy. This is because the quantity dx dy is infinitely small with respect to the other terms y dx and x dy: If we were to divide y dx, for example, and dx dy by dx we would find, on the one hand y, and on the other dy, which is the differential of the former, and is consequently infinitely smaller than it. From this it follows that the differential of the product of two quantities is equal to the product of the differential of the first of these two quantities with the second, plus the product of the differential of the second quantity with the first.

• 2. The differential of xyz is yz dx +xz dy +xy dz. This is because in considering the product xy as a single quantity, it is necessary, as we have just proven, to take the product of its differential y dx + x dy with the second quantity z (which gives yz dx +xz dy), plus the product of the differential dz [5] of the second quantity z by the first xy (which gives xy dz). Consequently the differential of xyz is seen to be yz dx +xz dy +xy dz.

• 3. The differential of xyzu is uyz dx +uxz dy +uxy dz +xyz du. This is proved as in the previous case, by considering xyz as a single quantity. It is similar for other cases to infinity,⁶ from which we form the following rule.

FormalPara Rule II.

 For multiplied quantities.

The differential of a product of several quantities multiplied together is equal to the sum of the products of the differential of each of the quantities multiplied by the product of the others.⁷

Thus, the differential of ax is x 0 + a dx, that is a dx. The differential of ⁸ $\overline{a + x} \times \overline{b - y}$ is b dx − y dx − a dy − x dy.

FormalPara Proposition III.

 Problem. (§6) To take the differential of any fraction. ⁹

The differential of $\frac{x} {y}$ is $\frac{y\,\mathit{dx}-x\,\mathit{dy}} {\mathit{yy}}$. This is because, if suppose that $\frac{x} {y} = z$, we have x = yz. Because these two variable quantities x and yz must always be equal to one another, whether increasing or decreasing, it follows that their differentials, that is to say their increments or decrements, are equal to each other. It then follows (see §5) that dx = y dz + z dy, and so

$$\displaystyle{\mathit{dz} = \frac{dx - z\,\mathit{dy}} {y} = \frac{y\,\mathit{dx} - x\,\mathit{dy}} {\mathit{yy}},}$$

substituting the value $\frac{x} {y}$ for z. This is what we wished to find, and we form the following rule.

FormalPara Rule III.

 For divided quantities, or fractions.

The differential of any fraction is equal to [6] the product of the differential of the numerator by the denominator minus the product of the differential of the denominator by the numerator all divided by the square of the denominator.

Thus, the differential of $\frac{a} {x}$ is $\frac{-a\,\mathit{dx}} {\mathit{xx}}$ and the differential of $\frac{x} {a+x}$ is $\frac{a\,\mathit{dx}} {\mathit{aa}+2\mathit{ax}+\mathit{xx}}$.

FormalPara Proposition IV.

 Problem. (§7) To take the differential of any power, whether perfect or imperfect, ¹⁰ of a variable quantity.¹¹

In order to give a general rule that applies to perfect and imperfect powers, it is necessary to explain the analogy that we find among their exponents.

If we consider a geometric progression¹² whose first term is unity and whose second term is any quantity x, and if under each term we write its exponent, it is clear that these exponents form an arithmetic progression.

Table 1

If we continue the geometric progression below unity and the arithmetic progression below zero, the terms of the one are the exponents of the corresponding terms in the other. Thus, − 1 is the exponent of $\frac{1} {x}$, − 2 that of $\frac{1} {\mathit{xx}}$, etc.

Table 2

If we introduce a new term in the geometric progression, we must introduce a similar term in the arithmetic progression to be its exponent.

Thus, $\sqrt{x}$ has $\frac{1} {2}$ as its exponent, $\root{3}\of{x}$ has $\frac{1} {3}$, $\root{5}\of{x^{4}}$ has $\frac{5} {4}$, $\frac{1} {\sqrt{x^{3}}}$ has $-\frac{3} {2}$, $\frac{1} {\root{3}\of{x^{5}}}$ has $-\frac{5} {3}$, $\frac{1} {\sqrt{x^{7}}}$ has $-\frac{7} {2}$, etc., so that the expressions [7] $\sqrt{x}$ and $x^{\frac{1} {2} }$, $\root{3}\of{x}$ and $x^{\frac{1} {3} }$, $\root{5}\of{x^{4}}$ and $x^{\frac{4} {5} }$, $\frac{1} {\sqrt{x^{3}}}$ and $x^{-\frac{3} {2} }$, etc., signify the same thing.

Table 3

Table 4

Table 5

Table 6

Table 7

Table 8

From this we see that just as $\sqrt{x}$ is the geometric mean between 1 and x, so also is $\frac{1} {2}$ the arithmetic mean between their exponents zero and 1, and just as $\root{3}\of{x}$ is the first of two means that are geometrically proportional between 1 and x, so also is $\frac{1} {3}$ the first of two means that are arithmetically proportional between their exponents zero and 1. It is the same for the others. Now, two things follow from the nature of these two progressions:

• 1. That the sum of the exponents of any two terms of the geometric progression is the exponent of the term that is their product. Thus, x ⁴⁺³ or x ⁷ is the product of x ³ by x ⁴, $x^{\frac{1} {2} +\frac{1} {3} }$ or $x^{\frac{5} {6} }$ is the product of $x^{\frac{1} {2} }$ by $x^{\frac{1} {3} }$, $x^{-\frac{1} {3} +\frac{1} {5} }$ or $x^{-\frac{2} {15} }$ is the product of $x^{-\frac{1} {3} }$ by $x^{\frac{1} {5} }$, etc. Similarly, $x^{\frac{1} {3} +\frac{1} {3} }$ or $x^{\frac{2} {3} }$ is the product of $x^{\frac{1} {3} }$ by itself, that is to say its square, x ⁺²⁺²⁺² or x ⁶ is the product of x ² by x ² by x ², that is to say its cube, $x^{-\frac{1} {3} -\frac{1} {3} -\frac{1} {3} -\frac{1} {3} }$ or $x^{-\frac{4} {3} }$ is the fourth power of $x^{-\frac{1} {3} }$, and so on for other powers. From this it is clear that the double, triple, etc., of the exponent of any term in the geometric progression is the exponent of the square, the cube, etc., of this term, and consequently that the half, the third, etc., of the exponent of any term of the geometric progression is the exponent of the square root, the cube root, etc., of this term;

• 2. That the difference of the exponents of any two terms of the geometric progression is the exponent of [8] the quotient of the division of these terms. Thus, $x^{\frac{1} {2} -\frac{1} {3} } = x^{\frac{1} {6} }$ is the exponent of the quotient of the division of $x^{\frac{1} {2} }$ by $x^{\frac{1} {3} }$, and $x^{-\frac{1} {3} -\frac{1} {4} } = x^{-\frac{7} {12} }$ is the exponent of the quotient of the division of $x^{-\frac{1} {3} }$ by $x^{\frac{1} {4} }$, from which we see that multiplying $x^{-\frac{1} {3} }$ by $x^{-\frac{1} {4} }$ is the same thing as dividing $x^{-\frac{1} {3} }$ by $x^{\frac{1} {4} }$. It is the same for the others.

This being well understood, there are two cases that may arise.

The first case is that of a perfect power, that is to say when the exponent is a whole number.¹³ The differential of xx is 2x dx, of x ³ is 3xx dx, of x ⁴ is 4x ³ dx, etc. Because the square of x is nothing but the product of x by x, its differential is x dx + x dx (see §5), that is to say 2x dx. Similarly, the cube of x being nothing but the product of x by x by x, its differential is xx dx +xx dx +xx dx (see §5), that is to say 3xx dx. However, because it is thus for other powers up to infinity, it follows that if we suppose that m indicates whatever whole number we might wish, the differential of x ^m is mx ^m−1 dx.

If the exponent is negative, we find that the differential of x ^−m or of $\frac{1} {x^{m}}$ is

$$\displaystyle{\frac{-\mathit{mx}^{m-1}\,\mathit{dx}} {x^{2m}} = -\mathit{mx}^{-m-1}\,\mathit{dx}.}$$

The second case is that of an imperfect power, that is to say when the exponent is a fractional number. Suppose we are to take the differential of $\root{n}\of{x^{m}}$, or $x^{\frac{m} {n} }$, where $\frac{m} {n}$ denotes any fractional number. We let $x^{\frac{m} {n} } = z$ and, raising both sides to the power n, we have x ^m = z ⁿ. Taking differentials as we have explained in the previous case, we find that mx ^m−1 dx = nz ⁿ⁻¹ dz, and so

$$\displaystyle{dz = \frac{\mathit{mx}^{m-1}\,\mathit{dx}} {\mathit{nz}^{n-1}} = \frac{m} {n} x^{\frac{m} {n} -1}\,\mathit{dx},}$$

or $\frac{m} {n} \,dx\,\root{n}\of{x^{m-n}}$, by substituting the value $nx^{m-\frac{m} {n} }$ in place of nz ⁿ⁻¹. If the exponent is negative, we find that the differential of $x^{-\frac{m} {n} }$ or $\frac{1} {x^{\frac{m} {n} }}$ is

$$\displaystyle{\frac{-\frac{m} {n} x^{\frac{m} {n} -1}\,\mathit{dx}} {x^{\frac{2m} {n} }} = -\frac{m} {n} x^{-\frac{m} {n} -1}\,\mathit{dx}.}$$

[9] This gives us the following general rule.

FormalPara Rule IV.

For powers, both perfect and imperfect.

The differential of any power, whether perfect or imperfect, of a variable quantity is equal to the product of the exponent of this power by the same quantity raised to a power diminished by unity, and multiplied by its differential.

Thus, if we suppose that m denotes either a whole or fractional number, whether positive or negative, and x denotes any variable quantity, the differential of x ^m is always mx ^m−1 dx.

FormalPara Examples.

The differential of the cube of ay −xx, that is to say of $\overline{\mathit{ay} -\mathit{xx}}^{\,3}$, is $3\times \overline{\mathit{ay} -\mathit{xx}}^{\,2}\times \overline{a\,\mathit{dy} - 2x\,\mathit{dx}} = 3a^{3}\mathit{yy}\,\mathit{dy}-6\mathit{aaxxy}\,\mathit{dy}+3\mathit{ax}^{4}\,\mathit{dy}-6\mathit{aayyx}\,\mathit{dx}+12\mathit{ayx}^{3}\,\mathit{dx}-6x^{5}\,\mathit{dx}$.

The differential of $\sqrt{\mathit{xy } + \mathit{yy}}$, or of $\overline{\mathit{xy} + \mathit{yy}}^{\,\frac{1} {2} }$, is

$$\displaystyle{\frac{1} {2} \times \overline{\mathit{xy} + \mathit{yy}}^{\,-\frac{1} {2} } \times \overline{y\,\mathit{dx} + x\,\mathit{dy} + 2y\,\mathit{dy}},}$$

or

$$\displaystyle{\frac{y\,\mathit{dx} + x\,\mathit{dy} + 2y\,\mathit{dy}} {2\sqrt{\mathit{xy } + \mathit{yy}}}.}$$

The differential of $\sqrt{a^{4 } + \mathit{axyy}}$, or of $\overline{a^{4} + \mathit{axyy}}^{\,\frac{1} {2} }$, is

$$\displaystyle{\frac{1} {2} \times \overline{a^{4} + \mathit{axyy}}^{\,-\frac{1} {2} } \times \overline{\mathit{ayy}\,\mathit{dx} + 2\mathit{axy}\,\mathit{dy}},}$$

or

$$\displaystyle{\frac{\mathit{ayy}\,\mathit{dx} + 2\mathit{axy}\,\mathit{dy}} {2\sqrt{a^{4 } + \mathit{axyy}}}.}$$

The differential of¹⁴ $\root{3}\of{\mathit{ax} + \mathit{xx}}$, or of $\overline{\mathit{ax} + \mathit{xx}}^{\frac{1} {3} }$, is

$$\displaystyle{\frac{1} {3} \times \overline{\mathit{ax} + \mathit{xx}}^{-\frac{2} {3} } \times \overline{a\,\mathit{dx} + 2x\,\mathit{dx}},}$$

or

$$\displaystyle{\frac{a\,\mathit{dx} + 2x\,\mathit{dx}} {3\root{3}\of{\mathit{ax} + \mathit{xx}}^{\;2}}.}$$

The differential of¹⁵ $\sqrt{ \mathit{ax} + \mathit{xx} + \sqrt{a^{4 } + \mathit{axyy}}}$, or of

$$\displaystyle{\overline{\mathit{ax} + \mathit{xx} + \sqrt{a^{4 } + \mathit{axyy}}}^{\frac{1} {2} },}$$

is

$$\displaystyle{\frac{1} {2} \times \overline{\mathit{ax} + \mathit{xx} + \sqrt{a^{4 } + \mathit{axyy}}}^{\;-\frac{1} {2} } \times \overline{a\,\mathit{dx} + 2x\,\mathit{dx} + \frac{\mathit{ayy}\,\mathit{dx}+2\mathit{axy}\,\mathit{dy}} {2\sqrt{a^{4 } +\mathit{axyy}}} },}$$

or

$$\displaystyle{ \frac{a\,\mathit{dx} + 2x\,\mathit{dx}} {2\sqrt{\mathit{ax } + \mathit{xx } + \sqrt{a^{4 } + \mathit{axyy}}}} + \frac{\mathit{ayy}\,\mathit{dx} + 2\mathit{axy}\,\mathit{dy}} {2\sqrt{a^{4 } + \mathit{axyy}} \times 2\sqrt{\mathit{ax } + \mathit{xx } + \sqrt{a^{4 } + \mathit{axyy}}}}.}$$

[10] According to this rule and the rule of fractions (see §7 and 6), the differential of¹⁶

$$\displaystyle{ \frac{\root{3}\of{\mathit{ax} + \mathit{xx}}} {\sqrt{\mathit{xy } + \mathit{yy}}}}$$

is¹⁷

$$\displaystyle{\frac{\frac{a\,\mathit{dx}+2x\,\mathit{dx}} {3\root{3}\of{\mathit{ax}+\mathit{xx}}^{\;2}} \times \sqrt{\mathit{xy } + \mathit{yy}} + \frac{-y\,\mathit{dx}-x\,\mathit{dy}-2y\,\mathit{dy}} {2\sqrt{\mathit{xy } +\mathit{yy}}} \times \root{3}\of{\mathit{ax} + \mathit{xx}}} {\mathit{xy} + \mathit{yy}}.}$$

FormalPara Remark.

(§8) It is appropriate to note carefully that in taking differentials, we always suppose that, as one of the variables x increases, the others, y, z, etc., also increase. That is to say, as x becomes x +dx, then y, z, etc., become y +dy, z +dz, etc. This is why if it happens that some of them increase while others decrease, it is necessary to regard the differentials as negative quantities with respect to those others which we suppose are increasing, and consequently to change the signs of the terms containing the differentials of those which are decreasing. Thus, if we suppose that x increases and y and z decrease, that is to say that x becomes x +dx and y and z become y −dy and z −dz, and if we wish to take the differential of the product xyz, it is necessary in the differential xy dz +xz dy +yz dx already found (see §5), to change the signs of the terms which contain dy and dz. This gives the differential yz dx −xy dz −xz dy that we wished to find.