Problem 25 - 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. $$g(x)=\frac{x}{x-1}$$

Step-by-Step Solution

Answer

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

Step 1: Set Up the Limit Expression

The given function is \(g(x) = \frac{x}{x-1}\). To find \(g'(x)\) using the derivative definition, substitute \(z\) for \(x\) in \(g(z)\) and place it in the limit formula: \[g'(x) = \lim_{z \rightarrow x} \frac{g(z) - g(x)}{z-x} = \lim_{z \rightarrow x} \frac{\frac{z}{z-1} - \frac{x}{x-1}}{z-x}\].

Step 2: Simplify the Numerator

Find a common denominator for the fractions in the numerator: \(g(z) - g(x) = \left(\frac{z(x-1) - x(z-1)}{(z-1)(x-1)}\right)\). Simplify this to: \[\frac{zx - z - xz + x}{(z-1)(x-1)} = \frac{x-z}{(z-1)(x-1)}\].

Step 3: Simplify the Whole Expression

Substitute the simplified numerator back into the expression for \(g'(x)\): \[g'(x) = \lim_{z \rightarrow x} \frac{x - z}{(z-1)(x-1)(z-x)}\]. Cancel the \((x-z)\) from the numerator and denominator: \[g'(x) = \lim_{z \rightarrow x} \frac{-1}{(z-1)(x-1)}\].

Step 4: Evaluate the Limit

Now that the expression is simplified, evaluate the limit as \(z\) approaches \(x\): since the term \((z-1)\) does not depend on \(x\): \[g'(x) = \frac{-1}{(x-1)^2}\].

Step 5: Finalize the Derivative

The derivative of \(g(x) = \frac{x}{x-1}\) is \(g'(x) = \frac{-1}{(x-1)^2}\). Verify this result for accuracy and consistency with derivative rules.

Key Concepts

Limit Definition of Derivative

The limit definition of a derivative is a fundamental concept in calculus. It provides the basis for calculating the derivative of a function at any given point and can be understood as the slope of the tangent line to the function at that point.

Let's break down the formula:

• We begin with a function, in this case, denoted as \(f(x)\).
• The limit definition is written as: \(f'(x) = \lim_{z \rightarrow x} \frac{f(z) - f(x)}{z-x}\).
• This formula calculates the difference in the function's values as \(z\) approaches \(x\), divided by the difference in \(z\) and \(x\) themselves.
• Essentially, you're looking at the average rate of change (or slope) as the gap between these two points approaches zero.

This approach is the start for derivative calculations and is essential for understanding how derivatives measure changes in functions.

Rational Functions

Rational functions are a type of function where the output is a quotient of two polynomials. For example, the function \(g(x) = \frac{x}{x-1}\) is a rational function.

Some properties of rational functions include:

• They have the form \(\frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomials.
• The domain of these functions excludes points where the denominator \(q(x)\) is zero, as they are undefined at these spots.
• They often have horizontal or vertical asymptotes, which are lines that the graph approaches but never touches.
• Studying the behavior of rational functions involves looking at their limits and asymptotes, often requiring algebraic manipulation to identify these properties.

Understanding rational functions is crucial when finding their derivatives, as seen in the example \(g(x)\). Rational functions, like all functions, can be differentiated using the limit definition of the derivative.

Derivative Rules

Derivative rules are shortcuts that help us quickly find derivatives without always using the limit definition. They make the process more efficient and include rules such as the power rule, product rule, quotient rule, and chain rule.

In the task:

• The derivative of our function, \(g(x) = \frac{x}{x-1}\), was found using a simplified version of the derivative rules.
• These shortcuts often rely on recognizing forms of functions and applying standard derivatives already known to derive the solution faster.
• Specifically, the derivative rules help in determining derivatives of more complex expressions quickly. This can include functions with multiple layers or ones that involve multiplication and division.
• However, in our case, creating the initial set-up required understanding the limit definition before applying algebraic manipulation to simplify.

Derivative rules are handy tools for tackling more complex derivatives and vastly reduce the time and computations needed to find these solutions.