Question

In Exercises, find the second derivative of the function. $$ f(x)=\frac{x+1}{x-1} $$

Short answer

The second derivative of \( f(x)=\frac{x+1}{x-1} \) is \( f''(x)=\frac{4}{(x-1)^3} \)

Step-by-step solution

Find the first derivative

To find the first derivative of the function \( f(x)=\frac{x+1}{x-1} \), use the quotient rule which states that the derivative of \( \frac{u}{v} \) is \( \frac{v\cdot u' - u\cdot v'}{v^2} \). Here, \( u = x+1 \), its derivative \( u' = 1 \), \( v = x-1 \), and its derivative \( v' = 1 \). Substituting these values gives \( f'(x)=\frac{(x-1) \cdot 1 - (x+1) \cdot 1}{(x-1)^2}=\frac{-2}{(x-1)^2} \)

Find the second derivative

Having obtained the first derivative as \( f'(x)=\frac{-2}{(x-1)^2} \), this can also be written as \( f'(x)=-2 \cdot (x-1)^{-2} \). Now to find the second derivative, we will use the power rule in differentiation which states that if \( g(x) = x^n \), then \( g'(x) = n \cdot x^{n-1} \). For \( f'(x)=-2 \cdot (x-1)^{-2} \), \( n=-2 \), and **u**=x-1. So \( f''(x) = -2 \cdot -2 \cdot (x-1)^{-3} \cdot 1 = \frac{4}{(x-1)^3} \).

Key concepts

Quotient Rule

Understanding the quotient rule is essential when dealing with the differentiation of functions that are formed by the division of two other functions. Imagine you are tasked with finding the rate of change or slope of a curve at any point for a function that is the ratio of two functions—this is where the quotient rule comes into play.

The quotient rule states that to differentiate a quotient such as \( \frac{u}{v} \), where \( u \) and \( v \) are both functions of \( x \), the derivative \( \frac{du}{dx} \) is given by the formula \( \frac{v\cdot u' - u\cdot v'}{v^2} \). To put it simply, the derivative is the difference of the product of the denominator and the derivative of the numerator and the product of the numerator and the derivative of the denominator, all divided by the square of the denominator.

In the exercise provided, \( f(x)=\frac{x+1}{x-1} \), we identify \( u \) as \( x+1 \) and \( v \) as \( x-1 \). Then with the derivatives \( u'=1 \) and \( v'=1 \), following the quotient rule, we find the first derivative \( f'(x) \) of the given function. This skill is particularly handy when working with rational functions and becomes second nature to students over time as they practice more examples.

Power Rule

The power rule is among the most fundamental tools in calculus for finding the derivative of a function. It is especially straightforward and highly practical for dealing with functions that include powers of \( x \).

The power rule simply states that if a function \( g(x) \) is given as \( x^n \) where \( n \) is any real number, then the derivative \( g'(x) \) is \( n\cdot x^{n-1} \). It's a useful shortcut that eliminates the need for the more complex limit definition of a derivative each time we want to find the rate of change of a function.

In the case of the second derivation of our example function \( f'(x) \) where we have \( f'(x)=-2\cdot(x-1)^{-2} \), the application of the power rule allows us to quickly determine the second derivative \( f''(x) \). As the exercise suggests, we identify \( n=-2 \) and apply the power rule to derive \( f''(x)=4\cdot(x-1)^{-3} \) which simplifies to \( \frac{4}{(x-1)^3} \). The power rule is a key player in the toolkit of calculus techniques, simplifying the process of differentiation and is vital for students to master early on.

Differentiation

Differentiation is the cornerstone of calculus, concerned with calculating the rate at which one quantity changes with respect to another. When a student understands differentiation, they can find the slopes of curves, determine the rate of change of one variable relative to another, and solve a variety of real-world problems involving rates of change, like those in physics and economics.

At its core, differentiation is about finding derivatives. In the context of our example, the derivative \( f'(x) \) of the function \( f(x) \) represents the rate of change of \( f \) with respect to \( x \). The second derivative \( f''(x) \) tells us about the acceleration, or the rate at which this rate of change itself is changing.

Students should embrace differentiation as more than just a set of rules and formulas. It's a way to understand and describe the world around us, from the paths of planets to the growth of a population. As they progress in their calculus studies, students will be able to apply differentiation to increasingly complex and abstract concepts, sharpening their mathematical intuition and problem-solving skills.