Question

Assume the derivatives of \(f\) and \(g\) exist. In this section, we showed that the rule \(\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}\) is valid for what values of \(n ?\)

Short answer

Based on the proof above, the valid values for n in the power rule of differentiation, \(\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}\), are all values of n except \(n=1\).

Step-by-step solution

Recall the definition of the derivative

The derivative of a function at a point x is defined as the limit: $$\frac{d}{d x}\left(x^{n}\right) = \lim_{h \to 0}\frac{(x+h)^n - x^n}{h}$$

Apply the binomial theorem

Apply the binomial theorem, which states that for any real number n, we have: $$(x + h)^n = \sum_{k=0}^n \dbinom{n}{k}x^{n-k}h^k$$ So, $$\frac{d}{d x}\left(x^{n}\right) = \lim_{h \to 0}\frac{\sum_{k=0}^n \dbinom{n}{k}x^{n-k}h^k - x^n}{h}$$

Simplify the expression

Simplify the expression and isolate the x^n term: $$\frac{d}{d x}\left(x^{n}\right) = \lim_{h \to 0}\frac{\dbinom{n}{0}x^{n}h^0 + \dbinom{n}{1}x^{n-1}h^1 + \sum_{k=2}^n \dbinom{n}{k}x^{n-k}h^k - x^n}{h}$$ Replace the binomial coefficients with their definitions, and note that \(\dbinom{n}{0} = 1\) and \(\dbinom{n}{1} = n\): $$\frac{d}{d x}\left(x^{n}\right) = \lim_{h \to 0}\frac{x^n(1-n) + nx^{n-1}h + h^2\sum_{k=2}^n \dbinom{n}{k}x^{n-k}h^{k-2}}{h}$$

Divide by h and take the limit as h approaches 0

Divide through by h and then take the limit as h approaches 0: $$\frac{d}{d x}\left(x^{n}\right) = \lim_{h \to 0}\left[(1-n)x^n + nx^{n-1} + h\sum_{k=2}^n \dbinom{n}{k}x^{n-k}h^{k-1}\right]$$ As h approaches 0, the third term (with the summation) goes to 0 since it contains h: $$\frac{d}{d x}\left(x^{n}\right) = (1-n)x^n + nx^{n-1}$$ Now, we want the left-hand side of this equation to be equal to the power rule result, i.e., \(nx^{n-1}\): $$\frac{d}{d x}\left(x^{n}\right) = nx^{n-1}$$

Determine the valid values of n

For this equality to hold, \((1-n)x^n = 0\), which is true for all x when \(n \neq 1\). Therefore, the power rule is valid for all values of n except \(n=1\).

Key concepts

Derivative Definition

The derivative of a function gives us the rate at which the function's output changes as its input changes. It is an essential concept in calculus. The formal definition of the derivative of a function \( f(x) \) at a point \( x \) is expressed through a limit:
\[\frac{d}{dx} f(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}.\]
In simpler terms, it measures how much \( f(x) \) will increase or decrease for a small change \( h \) around the point \( x \).

Key aspects to remember about the derivative definition include:

• The limit needs to exist for the derivative to be defined at that point.
• The concept can be visualized as the slope of the tangent line to the graph of \( f(x) \) at \( x \).
• Derivatives form the foundation for rules such as the power rule, which simplifies the computation of derivatives for polynomial functions.

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to a power. Specifically, it provides a formula to expand \((x + h)^n\) as a sum of individual terms involving coefficients, powers of \( x \), and powers of \( h \):
\[(x + h)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} h^k.\]
This expansion is crucial when dealing with the derivative of powers, especially in the context of polynomial functions where it assists in simplifying complex expressions.

Important points about the binomial theorem:

• \(\binom{n}{k}\) represents a binomial coefficient, indicating the number of ways to choose \( k \) elements from a set of \( n \) elements.
• The theorem holds for any real number \( n \), which is useful in calculus applications.
• Using the expansion, it becomes easier to observe the structure of expressions when a small change \( h \) is considered.

Differentiation of Powers

Differentiation of powers is a fundamental technique in calculus that allows us to easily find the derivative of functions of the form \( x^n \). The power rule simplifies this process and states that:
\[ \frac{d}{dx} x^n = nx^{n-1} \]
This rule is valid for nearly all real number values of \( n \), with the main requirement being \( n eq 1 \) as shown in the solution.

Crucial insights for differentiation of powers include:

• The power rule is derived from the definition of the derivative using limits and the binomial theorem.
• It effectively reduces the problem of differentiation to multiplying and reducing the power by one, making it quick and straightforward.
• Understanding how to differentiate powers is essential for solving more complex calculus problems involving polynomial and non-polynomial functions.

Knowing how to apply the power rule fundamentally supports calculus computations by providing a reliable method for dealing with variable exponents.