Problem 16 - A History of Mathematics: An Introduction

Question

Find the ratio of the fluxion of \(x\) to the fluxion of \(1 / x\) using Newton's "synthetic" method of fluxions.

Step-by-Step Solution

Answer

Answer: \(\frac{x^2}{-1}\)

Step 1: Find the derivative of x

Using Newton's method of fluxions, the derivative of \(x\) with respect to \(x\) (written as \(\dot{x}\)) is simply 1, because the rate of change of \(x\) with respect to itself is always 1.

Step 2: Find the derivative of \(\frac{1}{x}\)

To find the derivative of \(\frac{1}{x}\), we can rewrite the function as \(x^{-1}\). Using the power rule of differentiation, we find the derivative of \(x^{-1}\) as follows: \(\dot{(\frac{1}{x})} = -x^{-2} = -\frac{1}{x^2}\)

Step 3: Find the ratio of the two derivatives

Now that we have found the derivatives of \(x\) and \(\frac{1}{x}\), we can find the ratio of their derivatives as: \(\frac{\dot{x}}{\dot{(\frac{1}{x})}} = \frac{1}{-\frac{1}{x^2}} = \frac{x^2}{-1}\) Therefore, the ratio of the fluxion of \(x\) to the fluxion of \(\frac{1}{x}\) using Newton's "synthetic" method of fluxions is \(\boxed{\frac{x^2}{-1}}\).

Key Concepts

Newton's method

Newton's method of fluxions is a technique developed by Sir Isaac Newton as a way to describe motion and change. In simpler terms, it's an early method of calculus focusing on instantaneous rates of change. Instead of using terms like derivatives, Newton used the concept of fluxions. Fluxions represent how quickly these variables change over time. You can think of fluxions as early versions of our modern derivative notation.
This method allows us to calculate the derivative by considering how coordinates of a point change with respect to each other. In the context of the exercise, finding the fluxion of a simple variable like \(x\) means we're observing how \(x\) changes with itself, which leads to \(\dot{x} = 1\). Newton's method establishes foundations that led to modern differentiation techniques, helping us analyze functions and their rates of change easily.

Differentiation

Differentiation is the core process of calculus used to determine the rate at which a function is changing at any point. It involves calculating the derivative, which can be thought of as the "slope" of a function at a particular point. In mathematical terms, this is finding the limit of the ratio of the change in the function's output to the change in the input, as the change approaches zero.
When we differentiate a function, we produce another function, called the derivative. This derivative function gives us the slope of the original function at any given point. In the exercise, we calculated the derivative of \(x\) and \(\frac{1}{x}\) by finding their respective fluxions. For \(x\), the fluxion is 1, illustrating that its rate of increase relative to itself is always constant. Differentiating \(\frac{1}{x}\), rewritten as \(x^{-1}\), demonstrates using differentiation to find a function's changing rate concerning another variable.

Power rule

The power rule is a basic, yet powerful tool in calculus that simplifies finding derivatives. It tells us that if you have a function \(f(x) = x^n\), where \(n\) is any real number, then the derivative \(f'(x)\) is \(nx^{n-1}\). This rule drastically reduces the complexity of differentiating power functions, making it easier to find slopes swiftly.
Take our example with \(\frac{1}{x}\), written as \(x^{-1}\). By applying the power rule, we multiply by the exponent and subtract one from it: the derivative becomes \(-1 \cdot x^{-2}\), or \(-\frac{1}{x^2}\). This calculation shows how straightforward and efficient the power rule is for differentiation. By using this rule, algebraic manipulation of complex functions becomes less daunting, allowing us to focus on interpreting derivative meanings readily.