Introduction

Understanding the First Principle: The first principle, also known as the definition of the derivative, states that the derivative of a function at a point is the limit of the difference quotient as the change in (x) approaches zero. The difference quotient is given by (frac{f(x h) - f(x)}{h}) as (h to 0). In this article, we will use this principle to differentiate (y x^{1/x}) from the first principle.

Key Concept: Solving for the derivative of a function at a point using the first principle involves forming the difference quotient, simplifying the expression, and then taking the limit as (h to 0).

Solving for the Derivative of (x^{1/x}): Step-by-Step Guide

Let's consider the function (y x^{1/x}) and find its derivative using the first principle. We start by forming the difference quotient:

Forming the Difference Quotient

For the function (y x^{1/x}), the difference quotient is defined as:

[D_{x^a} frac{(x h)^{1/(x h)} - x^{1/x}}{h}]

Now, let's simplify this expression step by step:

Simplifying the Difference Quotient

Firstly, we rewrite the expression in a form that makes sense for (x a): (D_{x^a} frac{(x h) - x cdot frac{1}{(x h) - frac{1}{x}}}{h}).

Next, we simplify further:

[D_{x^a} frac{1 - frac{1}{x(x h)}}{1 - frac{1}{x}}]

Now, substitute (x a) into the resulting expression:

[x^{1/x} 1 - frac{1}{x^2}]

Calculating the Limit as (h to 0)

Now we need to take the limit as (h to 0). The expression simplifies to:

[frac{dy}{dx} lim_{h to 0} left(1 - frac{1}{x^2 (x h)}right)]

As (h to 0), the term (frac{1}{x^2 (x h)}) approaches (frac{1}{x^2}), and thus the derivative is:

[frac{dy}{dx} 1 - frac{1}{x^2}]

Additional Proofs: Derivative of (x^{-1}) and (left(frac{1}{x}right)^{-1})

For the sake of completeness, let's prove the derivative of (y x^{-1}) and (left(frac{1}{x}right)^{-1}) from the first principle.

Derivative of (x^{-1})

Consider (f(x) x^{-1}) and find (f'(x)) from first principles:

[f'(x) lim_{h to 0} frac{(x h)^{-1} - x^{-1}}{h}]

This can be simplified as:

[f'(x) lim_{h to 0} frac{frac{1}{x h} - frac{1}{x}}{h}]

Expanding the fraction in the numerator:

[f'(x) lim_{h to 0} frac{frac{x - (x h)}{x(x h)}}{h}]

Which further simplifies to:

[f'(x) lim_{h to 0} frac{-h}{h(x^2 hx)} lim_{h to 0} frac{-1}{x^2 hx}]

As (h to 0), the term (x^2 hx) approaches (x^2), and thus:

[f'(x) -frac{1}{x^2}]

Derivative of (left(frac{1}{x}right)^{-1})

Let (g(x) left(frac{1}{x}right)^{-1}) and find (g'(x)) from first principles:

First, rewrite (g(x) x), then:

[g'(x) lim_{h to 0} frac{(x h) - x}{h} lim_{h to 0} frac{h}{h} 1]

Thus, the derivative is:

[g'(x) 1]

Therefore, we have proved that:

[left(frac{1}{x}right)^{-1} x]

Conclusion

In this article, we have explored how to differentiate (y x^{1/x}) using the first principle method. We detailed the process of forming the difference quotient, simplifying the expression, and taking the limit as (h to 0). Additionally, we proved from the first principle that the derivative of (x^{-1}) is (-frac{1}{x^2}) and that (left(frac{1}{x}right)^{-1} x).

Keywords: differentiation, first principle, limit process, x^(1/x)