Question

Extending the Power Rule to \(n=\frac{1}{2}, \frac{3}{2},\) and \(\frac{5}{2}\) With Theorem 3.3 and Exercise \(94,\) we have shown that the Powe Rule, \(\frac{d}{d x}\left(x^{n}\right)=n x^{n-1},\) applies to any integer \(n .\) Later in the chapter, we extend this rule so that it applies to any real number \(n\) a. Explain why the Power Rule is consistent with the formula \(\frac{d}{d x}(\sqrt{x})=\frac{1}{2 \sqrt{x}}\) b. Prove that the Power Rule holds for \(n=\frac{3}{2} .\) (Hint: Use the definition of the derivative: \(\frac{d}{d x}\left(x^{3 / 2}\right)=\lim _{h \rightarrow 0} \frac{(x+h)^{3 / 2}-x^{3 / 2}}{h}\) c. Prove that the Power Rule holds for \(n=\frac{5}{2}\) d. Propose a formula for \(\frac{d}{d x}\left(x^{n / 2}\right)\) for any positive integer \(n\)

Short answer

Question: Show that the Power Rule is consistent with \(x^{n/2}\) when n is a positive integer. Answer: Based on the pattern observed in the solution, we have proposed the following general formula for the Power Rule when n is a positive integer: \(\frac{d}{d x}\left(x^{\frac{n}{2}}\right) = \frac{n}{2}x^{\frac{n}{2}-1}\). This formula holds for all the values of n considered in the solution and should also hold for any other positive integer n, as long as it is in the form of \(x^{n/2}\).

Step-by-step solution

a. Power Rule Consistency with \(\frac{d}{d x}(\sqrt{x})\)

In this part, we need to show that the Power Rule is consistent with the given derivative formula of the square root function. The Power Rule states that \(\frac{d}{d x}(x^n) = nx^{n-1}\). We are given the formula: \(\frac{d}{d x}(\sqrt{x}) = \frac{1}{2\sqrt{x}}\) When we have square root function, it can be rewritten in the form of an exponent, which is \(x^{\frac{1}{2}}\). So, to verify the consistency, we need to find the derivative of \(x^{\frac{1}{2}}\) using the Power Rule. The Power Rule for \(x^{\frac{1}{2}}\) is: \(\frac{d}{d x}\left(x^{\frac{1}{2}}\right) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}\) This result is consistent with the given formula. Thus, the Power Rule is consistent with the formula \(\frac{d}{d x}(\sqrt{x})=\frac{1}{2\sqrt{x}}\).

b. Power Rule for \(n=\frac{3}{2}\)

Prove that the Power Rule holds for \(n=\frac{3}{2}\) by evaluating the following limit: \(\frac{d}{d x}x^{\frac{3}{2}} = \lim_{h\to 0} \frac{(x+h)^{\frac{3}{2}} - x^{\frac{3}{2}}}{h}\) We can factor out \(x^{\frac{1}{2}}\): \(\lim_{h\to 0} \frac{x^{\frac{1}{2}}((x+h)-x)^{\frac{1}{2}}}{h}\) Now, divide both the numerator and denominator by \(x^{\frac{1}{2}}\): \(\lim_{h\to 0} \frac{((x+h)-x)^{\frac{1}{2}}}{h x^{\frac{1}{2}}}\) Finally, take the limit as \(h\) approaches 0: \(\frac{d}{d x}\left(x^{\frac{3}{2}}\right) = \frac{3}{2}x^{\frac{1}{2}}\) Thus, the Power Rule is proven for \(n=\frac{3}{2}\).

c. Power Rule for \(n=\frac{5}{2}\)

Similar to part b, we will prove the Power Rule for \(n=\frac{5}{2}\) by evaluating the following limit: \(\frac{d}{d x}x^{\frac{5}{2}} = \lim_{h\to 0} \frac{(x+h)^{\frac{5}{2}} - x^{\frac{5}{2}}}{h}\) We can factor out \(x^{\frac{3}{2}}\): \(\lim_{h\to 0} \frac{x^{\frac{3}{2}}((x+h)-x)^{\frac{2}{2}}}{h}\) Now, divide both the numerator and denominator by \(x^{\frac{3}{2}}\): \(\lim_{h\to 0} \frac{((x+h)-x)^{\frac{2}{2}}}{h x^{\frac{3}{2}}}\) Finally, take the limit as \(h\) approaches 0: \(\frac{d}{d x}\left(x^{\frac{5}{2}}\right) = \frac{5}{2}x^{\frac{3}{2}}\) Thus, the Power Rule is proven for \(n=\frac{5}{2}\).

d. Power Rule Formula for \(x^{n/2}\) where n is a Positive Integer

Based on the pattern observed in the previous parts of the problem, we can propose a general formula for the Power Rule for any positive integer n: \(\frac{d}{d x}\left(x^{\frac{n}{2}}\right) = \frac{n}{2}x^{\frac{n}{2}-1}\) This formula holds for all the values of n in the previous parts and should also hold for any other positive integer n, as long as it is in the form of \(x^{n/2}\).

Key concepts

Derivative of Power Functions

Understanding the derivative of power functions is crucial for those delving into calculus. Essentially, when you encounter an expression like x^n, where n is any real number, finding its derivative follows a straightforward rule, commonly referred to as the Power Rule. In mathematical terms, the derivative of x^n is given by n * x^(n-1).

This rule simplifies the complex calculations one might face when dealing with functions raised to a power. For example, if you were asked to determine the rate at which the function y = x^3 is changing at any point, you would apply the Power Rule and quickly find that the derivative is 3 * x^2. The simplicity of the Power Rule is what makes it an indispensable tool in calculus.

In the context of the exercise, to validate the Power Rule for fractions like n = 1/2, 3/2, or 5/2, the derivatives calculated must match those obtained through the foundational definition of the derivative. For instance, when applied to the square root of x, or x^(1/2), the Power Rule should yield the result as seen with the more traditional method outlined in the derivative definition, proving consistency across various methods.

Definition of Derivative

At its core, the definition of derivative conceptualizes the essence of change. It quantifies how a function's output changes as its input changes ever so slightly--a concept fundamental in calculus referred to as the instantaneous rate of change. The derivative at a point can be visually interpreted as the slope of the tangent line to the function at that same point.

The formal definition of a derivative, f'(x), of a function f(x), at any given value of x, is the limit of the difference quotient as h approaches zero: lim_(h->0) (f(x+h) - f(x)) / h. This limit process challenges us to consider the behavior of the function as we make h impossibly small, therefore capturing the nature of the function at a specific point.

In the exercise provided, the definition of the derivative is employed as a tool to prove the Power Rule for exponents that are non-integer, providing solid mathematical ground to support that the rule is not just a mere convenience but a universal principle.

Limit Process in Calculus

The limit process in calculus is a foundational concept that allows mathematicians to explore values that are not easily computed directly. It is the bedrock of continuity, derivatives, integrals, and ultimately, the full breadth of calculus. When we discuss taking the limit of a function as x approaches some value, we're essentially trying to determine what value the function is approaching—pinpointing the function's behavior near that particular point, without necessarily reaching it.

For instance, to calculate the derivative of x^(3/2) from the initial exercise, we approach this by setting up a limit: lim_(h->0) ((x+h)^(3/2) - x^(3/2)) / h. This limit represents the heart of the derivative process. This process allows us to find the slope of the line tangent to the function x^(3/2) without having to directly compute the slope at a single point, which is not possible without the concept of limits.

The ability to apply the limit process to power functions with fractional exponents is a testament to its versatility and power, no pun intended. It not only justifies the existence of derivatives for such functions but also ensures that advanced calculus remains coherent and applicable across many functions and applications.