Problem 27 - A History of Mathematics: An Introduction

Question

Derive the general power rule \(d\left(x^{n}\right)=n x^{n-1}\) for \(n\) a positive integer using differentials.

Step-by-Step Solution

Answer

#Answer# The derivative of the function \(x^n\) with respect to \(x\) is given by the general power rule: \(d(x^n) = nx^{n-1}\), where \(n\) is a positive integer.

Step 1: 1. Understand the concept of differentials

Differentials are used to represent the change in the value of a function with respect to changes in the independent variable. The differential of a function \(f(x)\) is given by \(df(x)\). We can write the differential of the function \(f(x) = x^n\) as \(d(x^n)\).

Step 2: 2. Use differentials to derive the power rule

Let's derive the power rule using differentials. Consider the function \(f(x) = x^n\). We need to find the differential of this function. Using the definition of differentials, we can write \(\Delta f = f'(x) \Delta x\). In our case, \(\Delta f = d(x^n)\). To differentiate the function \(f(x) = x^n\), we will first rewrite it in terms of the sum of its terms. We can rewrite \(f(x + \Delta x) - f(x)\) as \(\left[(x + \Delta x)^n - x^n\right]\) and expand it. Now, let's expand the binomial expression \((x + \Delta x)^n\) using the binomial theorem: \((x + \Delta x)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k}(\Delta x)^k\). By subtracting \(x^n\) from this expression, we get: \(\Delta f = \left[\sum_{k=0}^n \binom{n}{k} x^{n-k} (\Delta x)^k\right] - x^n\) Notice that this expression is equal to \(d(x^n)\), which is the change in \(x^n\). Divide this expression by \(\Delta x\) to find the derivative of \(x^n\): \(\lim_{\Delta x \to 0}\frac{d(x^n)}{\Delta x} = \lim_{\Delta x \to 0}\left[\frac{\sum_{k=0}^n \binom{n}{k} x^{n-k} (\Delta x)^k - x^n}{\Delta x}\right]\) Now, observe that the first term in the sum is \(x^n\), which will be canceled by the last term of the expression. Also note that when \(k = 1\), the binomial coefficient \(\binom{n}{1} = n\): \(\lim_{\Delta x \to 0}\left[\frac{- \sum_{k=2}^n \binom{n}{k} x^{n-k} (\Delta x)^k + nx^{n-1} (\Delta x)}{\Delta x}\right]\) In the limit as \(\Delta x \to 0\), all terms with \((\Delta x)^k, k \ge 2\) become zero. Thus we are left with the term containing \(k = 1\): \(d(x^n) = nx^{n-1}\)

Step 3: 3. Conclusion

Using differentials, we have derived the general power rule for finding the derivative of a function in the form \(x^n\), where \(n\) is a positive integer. The derivative is given by \(d(x^n) = nx^{n-1}\).

Key Concepts

Power Rule

The Power Rule is one of the simplest and most useful techniques in differential calculus. It allows us to find the derivative of functions that are polynomials, particularly those in the form of \(x^n\). The rule states that the derivative of \(x^n\) with respect to \(x\) is given by \(nx^{n-1}\). This means you multiply the power by the coefficient (which is 1 if the coefficient is not specified), then decrease the power by one.

To understand the power rule, let's go step by step:

• Consider the function \(f(x) = x^n\), where \(n\) is a positive integer.
• Applying the power rule, to find \(f'(x)\), we multiply the exponent by the front coefficient (leading coefficient) and subtract one from the exponent.
• This results in the function \(nx^{n-1}\), which is the derivative of the function \(x^n\).

This process makes finding derivatives of polynomial functions straightforward and efficient, which is why the power rule is a fundamental part of calculus education.

Binomial Theorem

The Binomial Theorem is a powerful algebraic tool that expands expressions in the form \((x + y)^n\) into a sum of terms. This theorem is essential in calculus for expanding expressions before taking derivatives, especially when using the power rule.

Here's a breakdown of how the Binomial Theorem works:

• In the expression \((x + \Delta x)^n\), the theorem helps expand this as \(\sum_{k=0}^{n} \binom{n}{k} x^{n-k} (\Delta x)^k\).
• This expansion is necessary when considering small changes in \(x\), as seen in expressions involving \(\Delta x\).
• The coefficients \(\binom{n}{k}\) are called binomial coefficients, derived from combinations and represent the different ways of picking \(k\) elements from \(n\) without regard to order.

Using this theorem in calculus, especially when deriving rules like the power rule, provides the algebraic structure needed to handle terms involving \(\Delta x\) and simplifies finding limits and differentials.

Derivatives

Derivatives are the cornerstone of differential calculus. They measure how a function changes as its input changes, allowing us to understand the rate of change of a quantity.

Key points about derivatives:

• Derivatives are represented as \(f'(x)\) and provide the slope of the function at any given point \(x\).
• The process of finding a derivative is called differentiation.
• It allows you to predict changes and understand behavior for complex functions by knowing how the changes occur for small variations of \(x\).
• In the power rule application, we see derivatives simplify problems by allowing quick calculations of the slope for polynomial functions.

Derivatives have vast applications from physics to economics, helping to model phenomena involving growth, decay, optimization, and many others.