Question

Leibniz's Proof of the Product Rule Here's how Leibniz explained the Product Rule in a letter to his colleague John Wallis: It is useful to consider quantities infinitely small such that when their ratio is sought, they may not be considered zero, but which are rejected as often as they occur with quantities incomparably greater. Thus if we have \(x+d x, d x\) is rejected. Similarly we cannot have \(x d x\) and \(d x d x\) standing together, as \(x d x\) is incomparably greater than \(d x d x\) . Hence if we are to differentiate \(u v,\) we write $$\begin{aligned} d(u v) &=(u+d u)(v+d v)-u v \\ &=u v+v d u+u d v+d u d v-u v \\ &=v d u+u d v \end{aligned}$$ Answer the following questions about Leibniz's proof. (a) What does Leibniz mean by a quantity being rejected? (b) What happened to dudv in the last step of Leibniz's proof? (c) Divide both sides of Leibniz's formula $$d(u v)=v d u+u d v$$ by the differential \(d x .\) What formula results? (d) Why would the critics of Leibniz's time have objected to dividing both sides of the equation by \(d x ?\) (e) Leibniz had a similar simple (but not-so-clean) proof of the Quotient Rule. Can you reconstruct it?

Short answer

(a) Leibniz referred to quantities being 'rejected' as infinitesimal values being disregarded when they were in association with much larger quantities. (b) In Leibniz's proof, \(dudv\) was rejected or removed in the last step because it's considered infinitely small, hence negligible. (c) The resulting formula post division by \(dx\) is: \(d(uv) / dx = v(du/dx) + u(dv/dx)\). (d) Critics would have objected to the notion of dividing by an infinitesimally small quantity due to lack of rigorous mathematical definition. (e) The formulation of the Quotient Rule with similar principles obtain, \(d(u/v) = (vdu - udv) / v^2\).

Step-by-step solution

Explanation of 'quantity being rejected'

An infinitely small quantity being rejected means these are ignored when compared to significantly larger quantities because their impact on the outcome is negligible. When these infinitesimal quantities are combined with much larger ones, they are considered to be close to zero and therefore 'rejected'.

Analysis of 'the disappearance of \(dudv\)'

In the last step of Leibniz's proof, \(dudv\) has been removed because when dealing with derivatives, the differential quantities like \(dudv\) are considered 'infinitely small' and negligible in comparison to the other terms (this is accordance with the principle of 'quantity being rejected').

Division by \(dx\)

When both sides of Leibniz's formula \(d(uv) = vdu + udv\) are divided by \(dx\), the following formula results: \(d(uv) / dx = v(du/dx) + u(dv/dx)\). This is a version of the Product Rule in differential calculus, a formula that gives the derivative of the product of two functions in terms of their individual derivatives and the original functions.

Reasoning against division by \(dx\)

Critics of Leibniz's time would have objected to dividing both sides of the equation by \(dx\) because the quantity \(dx\) represents an infinitesimally small change in x, which was not well-defined or rigorous from a mathematical perspective at that time. The concept of dividing by an 'infinitely small' quantity was controversial and not universally accepted.

Leibniz's Quotient Rule

A similar line of reasoning that Leibniz used for the Product Rule can be applied to deduce the Quotient Rule. For two given functions u(x) and v(x), the derivative of their quotient is given by: \(d(u/v) = (vdu - udv) / v^2\). The derivation of the Quotient Rule can be made by considering the differential of the quotient u/v and reducing the terms by rejecting the terms containing higher orders of differentials.

Key concepts

Infinitesimal Calculus

Infinitesimal calculus is a branch of mathematics that deals with the concept of an infinitely small quantity. Infinitesimals are the foundations of calculus, which Leibniz used to explain concepts like derivatives and integrals.

In the context of Leibniz's work, the term 'infinitesimal' refers to quantities so small that they are not zero, yet they are so tiny that they can be neglected when compared to larger quantities. This reasoning helps in analyzing the behavior of functions when values change by an infinitesimal amount.

When Leibniz states that a quantity is 'rejected,' he means that it is so small in comparison to other terms that it does not significantly affect the outcome of a calculation. This is a key principle in differential calculus, where minute changes are studied to understand the rate of change of functions.

Differential Calculus

Differential calculus is a major part of calculus that focuses on the concept of change. Specifically, it involves studying the rate at which quantities change: derivatives. The derivative represents how a function's output changes when its input changes by an infinitesimal amount.

Leibniz helped to develop the notation and formalism behind this branch of calculus, which allows mathematicians to calculate slopes of tangent lines, optimize functions, and understand the dynamics of moving objects among other applications.

Differential calculus involves various rules and theorems to compute the derivatives of functions. Two key rules formulated by Leibniz are the product rule and the quotient rule, which describe how to find derivatives of products and quotients of functions, respectively.

Product Rule

The product rule is a fundamental theorem in differential calculus for determining the derivative of the product of two functions. This rule states that the derivative of a product is the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function.

Mathematically, this is expressed as:
\[ \frac{d(uv)}{dx} = v\frac{du}{dx} + u\frac{dv}{dx} \]
Leibniz's original proof simplifies the calculation of the differential of the product by 'rejecting' the term \(dudv\) as it is considered infinitesimally small. This reflects the genius of Leibniz's work in using infinitesimals to simplify complex calculations while maintaining logical consistency in results.

Quotient Rule

Following the pattern of the product rule, the quotient rule is another key theorem in differential calculus. It describes how to find the derivative of a quotient, that is, the division of one function by another.

The quotient rule can be formulated as follows:
\[ \frac{d(\frac{u}{v})}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} \]
Leibniz's approach to the proof is analogous to the product rule; it involves considering the differential of the quotient \(\frac{u}{v}\) and simplifying by ignoring higher order differentials. Despite its simplicity, this application of infinitesimals was innovative for Leibniz's time and laid the groundwork for the modern use of limits in calculus.