Problem 58 - Single Variable Calculus

Question

Write the expression as a derivative of a function of \(x\). \(\lim _{h \rightarrow 0} \frac{\frac{1}{x+h}+\sqrt{x+h}-\frac{1}{x}-\sqrt{x}}{h}\)

Step-by-Step Solution

Answer

The given expression can be written as a derivative of the function \(f(x) = \frac{1}{x} + \sqrt{x}\), i.e., \(\lim _{h \rightarrow 0} \frac{\frac{1}{x+h}+\sqrt{x+h}-\frac{1}{x}-\sqrt{x}}{h} = f'(x)\).

Step 1: Format the expression as a difference quotient

The first step is to rewrite the given expression as a difference quotient. To accomplish this, we focus on the numerator of the expression. A difference quotient is of the form \(\frac{f(x+h) - f(x)}{h}\). Given expression: \[\lim _{h \rightarrow 0} \frac{\frac{1}{x+h}+\sqrt{x+h}-\frac{1}{x}-\sqrt{x}}{h}\] Let \(f(x) = \frac{1}{x} + \sqrt{x}\). Then the expression becomes: \[\lim _{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}\]

Step 2: Calculate the derivative of the function \(f(x)\) using the limit definition of derivatives

Recall that the derivative of a function is given by: \[f'(x) = \lim _{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} \] Now that we have the expression in the form of the limit definition of a derivative, we can directly see that the expression represents the derivative of the function \(f(x) = \frac{1}{x} + \sqrt{x}\). Hence, the expression can be written as a derivative of \(f(x)\): \[\lim _{h \rightarrow 0} \frac{\frac{1}{x+h}+\sqrt{x+h}-\frac{1}{x}-\sqrt{x}}{h} = f'(x)\] Where the function is: \[f(x) = \frac{1}{x} + \sqrt{x}\]

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus that helps illustrate how a function changes. It is essentially the backbone of derivative definitions and calculations.
The core idea behind the difference quotient is to measure the rate at which a function changes with respect to changes in its input. Mathematically, it's represented as:

• \( \frac{f(x+h) - f(x)}{h} \)

This expression tells us how much the function \( f(x) \) changes when we slightly alter \( x \) by a small amount \( h \).
This quotient forms the basis for calculating derivatives, as it approximates the slope of the function at a point as \( h \) approaches zero. This concept is central in understanding instantaneous rates of change in various mathematical functions.

Derivative Calculation

Calculating derivatives involves using the limit of the difference quotient as \( h \) approaches zero. This process helps in determining the instantaneous rate of change of a function, which is crucial in many scientific and engineering applications.
To calculate the derivative of \( f(x) \) using limits, you start with:

• \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

This definition essentially tells you how the function \( f(x) \) behaves at a very small scale around \( x \). By applying this to specific functions and evaluating the limit, you can find precise derivatives.
Derivatives provide insights into the behavior of functions, like their slopes, increasing or decreasing trends, and convexity or concavity. Calculating derivatives often involves algebraic simplifications and limit evaluations to determine their exact values.

Derivative of a Function

The derivative of a function at a specific point provides the slope of the tangent line to the curve at that point. This is a powerful tool in calculus, allowing us to understand and predict the behavior of functions.
For the function \( f(x) = \frac{1}{x} + \sqrt{x} \), the derivative \( f'(x) \) can be interpreted as the rate at which \( f(x) \) changes as \( x \) changes. This concept is practical in real-world applications, from calculating velocity to understanding growth rates in various phenomena.
In the original exercise, the task was to express a complex limit as a derivative, using the function \( f(x) \). By recognizing the expression as analogous to the limit definition of a derivative, it becomes apparent that the derivative \( f'(x) \) gives a concise representation of the function's behavior near \( x \). Understanding how to derive and apply this knowledge is essential for deeper mathematical comprehension and problem-solving.