Question

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\frac{2}{x-2} $$

Short answer

The most efficient rule to find the derivative of the function \(f(x)=\frac{2}{x-2}\) is (d) Quotient Rule

Step-by-step solution

Recognizing the function

Look at the function \(f(x)=\frac{2}{x-2}\). This is a rational function, where the numerator function is constant and the denominator is a simple linear function.

Identifying the possible derivative rule

Let's go through each rule. \ (a) Simple Power Rule: This rule is used when we have function of the form \(x^n\) where n is any real number, but here we don't have any such power of x.\ (b) Constant Rule: This rule is typically used for constant functions, which is not the case here.\ (c) General Power Rule: This rule is for any function representing \(u(x)^n\), where u(x) is a function of x and n is any real number. Again, our function does not fit this criterion.\ (d)Quotient Rule: This rule is applied on a function of the form \(\frac{f(x)}{g(x)}\), where both f(x) and g(x) are differentiable functions of x. Our function fits into this category. It’s a quotient of two functions 2 and (x-2). So this is the rule applied to find the derivative of our function.

Solution

The rule which could be used most efficiently to find the derivative of the function \(f(x)=\frac{2}{x-2}\) is (d) Quotient Rule

Key concepts

Quotient Rule

The Quotient Rule is an essential tool for differentiating functions that are expressed as one function divided by another. When dealing with a function of the form \( \frac{f(x)}{g(x)} \), the Quotient Rule helps you find the derivative efficiently. This rule states that the derivative of the quotient of two functions is given by:\[\left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}.\]This formula might look a bit daunting at first, but it's straightforward once you break it down:

• \( f'(x) \): First, find the derivative of the numerator function \( f(x) \).
• \( g(x) \) and \( g'(x) \): Next, find the derivative of the denominator function \( g(x) \).
• Substitute these derivatives into the Quotient Rule formula.

This rule is particularly useful for functions like \( f(x)=\frac{2}{x-2} \), where you have a non-zero constant in the numerator and a simple expression in the denominator.

Rational Functions

Rational functions are expressions of the form \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials. These functions are common in calculus and require specific techniques for differentiation. A key feature of rational functions is that their graphs may have features like vertical or horizontal asymptotes, which arise from the behavior of these polynomials.For rational functions like \( f(x)=\frac{2}{x-2} \), the numerator is a constant, and the denominator is a linear polynomial. This specific structure affects the behavior of the function, especially near points where the denominator is zero, like \( x=2 \) in this example, leading to vertical asymptotes.Understanding these characteristics helps when applying differentiation techniques, such as the Quotient Rule, as you navigate defining domains and potential discontinuities.

Differentiation Techniques

Differentiation techniques are methods used to find the derivative of a function, which represents the rate of change of the function with respect to a variable. Knowing when and how to apply certain rules is crucial for efficiency and accuracy.Some common techniques include:

• Power Rule: Used for functions of the form \( x^n \), where you bring the power down and decrease the exponent by one.
• Product Rule: Applied to functions that are the product of two functions.
• Quotient Rule: Used when you have a division of two functions, as discussed earlier.
• Chain Rule: Utilized for composite functions, where one function is nested within another.

Practicing these techniques and understanding when to apply each one will greatly aid in differentiating more complex functions and tackling calculus problems efficiently.