Problem 23 - University Calculus: Early Transcendentals

Question

Use the formula $$f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$$ to find the derivative of the functions. $$f(x)=\frac{1}{x+2}$$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \frac{1}{x+2} \) is \( f'(x) = \frac{-1}{(x+2)^2} \).

Step 1: Understanding the Problem

We need to find the derivative of the function \( f(x) = \frac{1}{x+2} \) using the definition of the derivative. The definition is given as \( f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x} \). This will involve substituting \( f(x) \) into the formula and simplifying.

Step 2: Substitute into the Derivative Formula

Substitute \( f(z) = \frac{1}{z+2} \) and \( f(x) = \frac{1}{x+2} \) into the derivative formula: \[f'(x) = \lim_{z \to x} \frac{\frac{1}{z+2} - \frac{1}{x+2}}{z-x}\]

Step 3: Combine the Fractions

In order to simplify \( \frac{1}{z+2} - \frac{1}{x+2} \), find a common denominator: \[\frac{1}{z+2} - \frac{1}{x+2} = \frac{x+2 - (z+2)}{(z+2)(x+2)} = \frac{x - z}{(z+2)(x+2)}\]

Step 4: Simplify the Expression

Substitute the combined fractions into the limit expression: \[f'(x) = \lim_{z \to x} \frac{\frac{x-z}{(z+2)(x+2)}}{z-x}\] This simplifies to: \[f'(x) = \lim_{z \to x} \frac{x-z}{(z-x)(z+2)(x+2)}\] Since \( x-z = -(z-x) \), substitute and simplify: \[f'(x) = \lim_{z \to x} \frac{-(z-x)}{(z-x)(z+2)(x+2)} = \lim_{z \to x} \frac{-1}{(z+2)(x+2)}\]

Step 5: Evaluate the Limit

Now, evaluate the limit as \( z \to x \): \[f'(x) = \frac{-1}{(x+2)^2}\] Hence, the derivative of the function \( f(x) = \frac{1}{x+2} \) is \( f'(x) = \frac{-1}{(x+2)^2} \).

Key Concepts

Derivative

The concept of a derivative is fundamental in calculus as it represents the rate at which a function is changing at any given point. Simply put, the derivative gives us the slope of the tangent line to the curve of the function at a specific point. This helps in understanding how a function behaves as its input changes.

• The process of finding a derivative is called differentiation.
• The derivative of a function is often denoted by \( f'(x) \) or \( \frac{df}{dx} \).
• It is useful in a wide array of applications, such as physics, economics, and engineering, to model change and optimize solutions.

The derivative's formula \( f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x} \), known as the limit definition, captures how we evaluate it inherently. By computing the ratio of the small change in the function value \((f(z) - f(x))\) to the small change in \(x\) \((z - x)\), and taking the limit as \(z\) approaches \(x\), we capture the instantaneous rate of change at that point.

Limit

Limits are used to find the value that a function approaches as the input approaches some point. They play a crucial role in defining derivatives and integrals in calculus. Understanding limits is essential when dealing with continuous functions and calculating them accurately is key to uncovering the behavior of functions at specific points.

• Limits help in understanding the behavior of functions as inputs become very small or very large.
• The notation \(\lim_{z \to x} f(z) = L\) means that as \(z\) approaches \(x\), \(f(z)\) approaches \(L\).
• In derivatives, limits help us determine the slope of a curve at a point by seeing how the function \(f(z)\) changes as \(z\) gets infinitely close to \(x\).

When evaluating the limit in the original problem, it involves simplifying the expression to a form that can resolve as \(z\) heads towards \(x\), helping find the precise slope of the function \(f(x) = \frac{1}{x+2}\) at any given \(x\).

Rational Functions

Rational functions are quotients of two polynomials, presenting situations where the output drastically changes with the input. Typical features include vertical asymptotes, where the function value approaches infinity, and horizontal asymptotes showing how the function behaves as input becomes very large or very small.

• A rational function is expressed as \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials.
• The function \(f(x) = \frac{1}{x+2}\) is a simple rational function with a single point of discontinuity at \(x = -2\), where it becomes undefined.
• In calculus, rational functions are easier to differentiate directly or by using algebraic manipulation to simplify them before applying the derivative definition.

Understanding rational functions involves identifying and working through these points of discontinuity or undefined behavior, which are often critical in applying calculus concepts like derivatives to explore functions' properties and sketch graphs accurately.