Problem 51 - Calculus an Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\frac{1}{x-2} $$

Step-by-Step Solution

Answer

The derivative of the function \(y=\frac{1}{x-2}\) is \(y'=\frac{-1}{(x-2)^2}\). The reciprocal rule was used to find this derivative.

Step 1: Identify the Function Type

The function \(y=\frac{1}{x-2}\) is a rational function. Finding the derivative of a rational function typically involves using quotient rule. But since the numerator is 1, it would be more convenient to use the rule of the derivative for reciprocal functions.

Step 2: Differentiation using the Rule for Reciprocal Functions

Take the reciprocal of \(x-2\) which gives \(y=(x-2)^{-1}\). The rule for derivative for reciprocal functions is \((u^n)'=n*u^{n-1}*u'\), where \(u=x-2\), \(n=-1\), and \(u'=1\) for this case. So the derivative becomes \((-1)*(x-2)^{-2}*1\).

Step 3: Simplify the Derivative

Simplify the derivative to get \(y'=\frac{-1}{(x-2)^2}\).

Key Concepts

Differentiation Rules

The rules for finding the derivative of a function, known as differentiation rules, are essential for students to learn and apply. Differentiation, in mathematical terms, is the process of determining the rate at which a function is changing at any given point. There are several rules to remember, such as the power rule, product rule, quotient rule, and chain rule.

For example, the power rule states that for any function in the form of \(f(x)=x^n\), where \(n\) is any real number, the derivative is given by \(f'(x)=nx^{n-1}\). This rule is integral for dealing with polynomial functions. By understanding and applying these rules correctly, students can find derivatives for a broad range of functions efficiently.

Reciprocal Functions

Reciprocal functions are specific types of rational functions where the numerator is 1, and hence, are given by the form \( y = \frac{1}{u(x)} \), where \(u(x)\) is a polynomial. Understanding how to differentiate reciprocal functions is particularly useful because they often appear in calculus problems.

The derivative of a reciprocal function can be found using the rule \((u^{-1})' = -u^{-2} \times u'\). This is derived from combining the power rule with the chain rule, another differentiation strategy where functions are composed of other functions. When you have a simple reciprocal function like the one given in the exercise, recognizing this form can dramatically simplify the process of finding its derivative.

Rational Functions

Rational functions are quotients of two polynomials, generally taking the form \( y = \frac{p(x)}{q(x)} \). The functions are called rational because they represent ratios of polynomials. A special case of rational functions are reciprocal functions, where the numerator is a constant, typically 1.

Finding the derivative of a rational function can sometimes involve the quotient rule, but shortcuts exist for particular types of rational functions like reciprocal functions, as seen in the original exercise. It is crucial to identify when these shortcuts apply, as they can greatly simplify the process of differentiation, making it easier to arrive at the correct derivative with less computation.

Quotient Rule

The quotient rule is a technique used in calculus when differentiating rational functions where one function is divided by another. It is articulated as \( \left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \), where \( f(x) \) and \( g(x) \) are differentiable functions.

The quotient rule is especially useful when both the numerator and denominator are complex polynomials. However, in cases like our original exercise where the numerator is 1, using the reciprocal rule is a more efficient choice. Understanding when to use the quotient rule and when to opt for simpler rules not only saves time but also reduces potential errors in calculation.