Problem 17 - Calculus of a Single Variable

Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=\frac{1}{x-2} $$

Step-by-Step Solution

Answer

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

Step 1: Write Down the Function

First, write down the function that needs to be differentiated. In this case, the function is \(y= \frac{1}{x-2}\). Here \(f(x) = 1\) and \(g(x) = x-2\).

Step 2: Find the Derivative of the Function

The next step is to use the quotient rule to find the derivative of the function. The formula is:\[y' = \frac{f'(x)g(x) - f(x)g'(x)}{{g(x)}^2}\]Therefore,\(y' = \frac{0*(x-2) - 1*1}{{(x-2)}^2} = \frac{-1}{{(x-2)}^2}\).

Step 3: Simplify the Resulting Expression

After substitution, the resulting equation is \(y'\ = - \frac{1}{{(x-2)}^2}\). In this case, this is the simplest form of the derivative, so no further simplification is needed.

Key Concepts

Quotient Rule

The quotient rule is a method in calculus for finding the derivative of a function that is the ratio of two differentiable functions. It's akin to a formulaic dance where one must meticulously follow the steps to reach the correct rhythm of the resulting derivative. When we have a function expressed as \( y = \frac{f(x)}{g(x)} \), the quotient rule tells us that the derivative \( y' \) is given by:\[ y' = \frac{f'(x)g(x) - f(x)g'(x)}{{g(x)}^2} \]In our exercise, where \( y = \frac{1}{x-2} \), we identify \( f(x) = 1 \) and \( g(x) = x-2 \). Applying the quotient rule simplifies finding the derivative, even when the numerator and denominator are more complex. It's crucial for students to remember to square the denominator \( g(x) \) in the final formula and to carefully apply the derivative to both the top (numerator) and bottom (denominator) functions individually.

Differentiation

Differentiation is the process of finding the derivative of a function. Like solving an intricate puzzle, it involves understanding the rules and the specific techniques to apply when faced with various types of functions. While our focus in this article is on the quotient rule, differentiation encompasses several other rules and methods, such as the power rule, product rule, chain rule, and more. Each rule serves as a powerful tool on its own, allowing us to handle different calculus problems effectively. The derivative represents the rate at which a function's output value is changing at any given point and is foundational for many applications across sciences and engineering.

Calculus Problems

Calculus problems vary in complexity from simple operations of differentiation to intricate integrals and their applications. When tackling such problems, it's essential to have a systematic approach:

• Identify the type of problem and the rule required.
• Apply the rule carefully by breaking down the function, if necessary.
• Simplify the derivative to its most reduced form.

By doing so, students can navigate through the maze of calculus with direction and clarity. In the given exercise, we identified the necessity of the quotient rule, applied it methodically, and reached the simplest form of the derivative.

Derivative of a Function

The derivative of a function is a core concept in calculus representing the sensitivity to change of a quantity, which is dependent on another quantity. It serves as a mathematical telescope that allows us to zoom in on the behavior of functions at the infinitesimal level. Finding the derivative, as showcased in our exercise, often requires careful application of relevant rules. Derivatives hold immense importance in various fields such as physics, where they can represent velocity or acceleration; in economics, they might denote marginal costs or revenues; and in biology, rates of reaction. As we have unearthed in this exercise by finding \( y' = - \frac{1}{{(x-2)}^2} \), the derivative of a function helps us understand the rate of change at any given point on the curve.