Problem 11 - Single Variable Calculus

Question

Use the definition of the derivative to find the derivative of the function. What is its domain? \(f(x)=\frac{1}{x+2}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = \frac{1}{x+2}\) is \(f'(x) = \frac{1}{(x+2)^2}\). The domain of the function is \((-\infty, -2) \cup (-2, \infty)\).

Step 1: Recalling the definition of the derivative

The definition of the derivative of a function \(f(x)\) at position \(x\) is given by: \[f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}\] We have to use this definition to find the derivative of the given function \(f(x) = \frac{1}{x+2}\).

Step 2: Evaluating the difference quotient

First, we evaluate the difference quotient: \[\frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{(x+h)+2} - \frac{1}{x+2}}{h}\]

Step 3: Simplifying the difference quotient

Next, we simplify the numerator by finding a common denominator: \[\frac{(x+2) - [(x+h)+2]}{(x+2)[(x+h)+2]h} = \frac{x + 2 - x - h - 2}{(x+2)[(x+h)+2]h} = \frac{-h}{(x+2)(x+h+2)h}\] Now, we can simplify the difference quotient by canceling out the -h term in the numerator and denominator: \[\frac{1}{(x+2)(x+h+2)}\]

Step 4: Applying the limit

Now, we apply the limit of the difference quotient as h approaches 0: \[f'(x) = \lim_{h\to 0} \frac{1}{(x+2)(x+h+2)}\] As h approaches 0, the second term in the denominator becomes (x + 2): \[f'(x) = \frac{1}{(x+2)(x+2)} = \frac{1}{(x+2)^2}\]

Step 5: Finding the domain

The function is \(f(x) = \frac{1}{x+2}\). The domain of a rational function is all real numbers except for the values that make the denominator equal to zero. To find the domain, we set the denominator to zero and solve for x: \(x + 2 = 0\) \(x = -2\) The domain of the function is all real numbers except for -2. In interval notation, the domain is: \((-\infty, -2) \cup (-2, \infty)\) So, the derivative of the function is \(f'(x) = \frac{1}{(x+2)^2}\), and the domain of the function is \((-\infty, -2) \cup (-2, \infty)\).

Key Concepts

Difference Quotient

The difference quotient is a foundational concept for understanding derivatives. This quotient is a formula used to approximate the slope of the tangent line to a curve at a specific point, giving insight into the function's behavior at that point. The formula is \[\frac{f(x+h) - f(x)}{h}\]where \( h \) is a small increment in \( x \), and \( f(x) \) is the function value at \( x \).

• This formula calculates the average rate of change of the function over the interval \([x, x+h]\).
• It forms the backbone of the derivative as it gradually becomes the instantaneous rate of change when \( h \) approaches zero.

By simplifying the difference quotient and taking the limit as \( h \) tends to zero, we derive the derivative, which represents the slope of the tangent to the curve at any given point.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The function \( f(x) = \frac{1}{x+2} \) is a simple example of a rational function, where the numerator is a constant polynomial (which is 1), and the denominator is the linear polynomial \( x+2 \).

• These functions are defined at all points where the denominator is not zero.
• The behavior of rational functions often includes asymptotes and discontinuities at points where their denominator equals zero.

Understanding the nature of rational functions is crucial since they frequently appear in calculus. Their properties, such as domains and limits, must often be considered to solve various problems effectively.

Domain of a Function

The domain of a function defines the set of all possible input values (\( x \)-values) for which the function is defined. For rational functions, such as \( f(x) = \frac{1}{x+2} \), the domain includes all real numbers except those that make the denominator zero, because division by zero is undefined.

• In this exercise, to find the domain, we solve \( x+2=0 \) to get \( x=-2 \).
• Thus, the domain of \( f(x) \) is all real numbers except \( x = -2 \).

In interval notation, this is expressed as \((-\infty, -2) \cup (-2, \infty)\). Recognizing and determining the domain is important because it tells us where the function behaves as expected and helps avoid any undefined operations.

Limit Process

The limit process is a key tool in calculus used to find the derivative of a function. It involves resolving what happens to a function as the variable approaches a particular value. Specifically, for derivatives, this involves evaluating\[\lim_{h\to 0} \frac{f(x+h) - f(x)}{h}\]

• This process enables us to determine the instantaneous rate of change or the slope of the tangent at a specific point.
• By applying the limit as \( h \) approaches zero, the difference quotient essentially "zooms in" on a very small interval around \( x \), giving an accurate slope of the curve at that point.

Understanding the limit process is fundamental for calculating derivatives, making it an essential concept not just in calculus, but in various applications involving change or motion.