Find The Derivative Of F(x) = (x^2 - 2) / (x - 1)

Dec 17, 2025 by Alex Johnson 50 views

Welcome to the exciting world of calculus! Today, we're going to tackle a common calculus problem: finding the derivative of a function. Specifically, we'll be working with the function $f(x)= rac{x^2-2}{x-1}$ . Derivatives are fundamental to understanding how functions change, and they have applications in countless fields, from physics and engineering to economics and biology. So, let's dive in and learn how to find $f'(x)$ !

Understanding Derivatives: What's the Big Deal?

Before we get our hands dirty with the calculation, it's important to understand what a derivative actually represents. In simple terms, the derivative of a function at a particular point tells us the instantaneous rate of change of that function at that point. Think of it like looking at the speedometer in your car. The speedometer doesn't tell you your average speed over your entire trip; it tells you your speed right now. That's the essence of a derivative! It's the slope of the tangent line to the function's graph at a specific point. This concept is crucial for optimization problems (finding maximums and minimums), analyzing motion, and modeling complex systems. The process of finding a derivative is called differentiation, and it's a core skill for any budding mathematician or scientist.

The Quotient Rule: Our Powerful Tool

When we encounter a function that is a fraction, like our $f(x)= rac{x^2-2}{x-1}$ , we often need to use a specific rule for differentiation called the Quotient Rule. Trying to differentiate this function directly using the definition of the derivative (the limit definition) would be quite cumbersome. The Quotient Rule provides a much more elegant and efficient way to handle such situations. The rule states that if you have a function $f(x)$ that can be expressed as the ratio of two other functions, say $f(x) = rac{g(x)}{h(x)}$ , then its derivative $f'(x)$ is given by:

f'(x) = rac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}

Let's break this down. Here, $g(x)$ is the numerator of your original function, and $h(x)$ is the denominator. We need to find the derivative of the numerator ( $g'(x)$ ) and the derivative of the denominator ( $h'(x)$ ). Then, we plug these into the formula. It's like a recipe: multiply the derivative of the top by the bottom, subtract the top multiplied by the derivative of the bottom, and then divide all of that by the bottom squared. While it might seem a bit complex at first, with a little practice, the Quotient Rule becomes second nature. It's a cornerstone of differential calculus and indispensable for solving many calculus problems involving rational functions. Remember, mastering the Quotient Rule is key to efficiently differentiating fractions of functions, a common scenario in calculus.

Step-by-Step Differentiation of $f(x)= rac{x^2-2}{x-1}$

Now, let's apply the Quotient Rule to our specific function, $f(x)= rac{x^2-2}{x-1}$ .

Identify $g(x)$ and $h(x)$ : In our function, the numerator is $g(x) = x^2 - 2$ . The denominator is $h(x) = x - 1$ .
Find the derivatives $g'(x)$ and $h'(x)$ : To find $g'(x)$ , we differentiate $g(x) = x^2 - 2$ . Using the power rule (which states that the derivative of $x^n$ is $nx^{n-1}$ ), the derivative of $x^2$ is $2x^{2-1} = 2x$ . The derivative of a constant (like -2) is always 0. So, $g'(x) = 2x - 0 = 2x$ . To find $h'(x)$ , we differentiate $h(x) = x - 1$ . The derivative of $x$ (which is $x^1$ ) is $1x^{1-1} = 1x^0 = 1$ . The derivative of the constant -1 is 0. So, $h'(x) = 1 - 0 = 1$ .
Plug into the Quotient Rule formula: Now we substitute our findings into the Quotient Rule: $f'(x) = rac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$ .
$f'(x) = rac{(2x)(x-1) - (x^2-2)(1)}{(x-1)^2}$
Simplify the numerator: Let's expand and simplify the numerator:
- $(2x)(x-1) = 2x^2 - 2x$
- $(x^2-2)(1) = x^2 - 2$ So, the numerator becomes: $(2x^2 - 2x) - (x^2 - 2)$ . Be careful with the subtraction here; the negative sign applies to both terms in the second parenthesis. $2x^2 - 2x - x^2 + 2$ Combine like terms: $(2x^2 - x^2) - 2x + 2 = x^2 - 2x + 2$ .
Write the final derivative: Now, we put the simplified numerator back over the squared denominator:
$f'(x) = rac{x^2 - 2x + 2}{(x-1)^2}$

And there you have it! The derivative of $f(x)= rac{x^2-2}{x-1}$ is $f'(x) = rac{x^2 - 2x + 2}{(x-1)^2}$ . Congratulations, you've successfully navigated the Quotient Rule! This process demonstrates how to systematically break down a problem into smaller, manageable steps, a valuable skill not just in math but in life.

Why is This Important? Applications of Derivatives

So, we've found the derivative, $f'(x)$ , but why does this matter? Understanding the derivative of $f(x)= rac{x^2-2}{x-1}$ has practical implications. For instance, this specific function might represent a cost function, a rate of production, or a velocity in a physical scenario. By finding its derivative, we can determine the marginal cost (the cost of producing one more unit), the instantaneous rate of change of production, or the velocity at any given time. In optimization problems, setting the derivative equal to zero can help us find critical points where the function might reach a maximum or minimum value. Imagine a company trying to maximize its profit; they would use derivatives to find the production level that yields the highest profit. In physics, if $f(x)$ represented the position of an object over time, then $f'(x)$ would represent its velocity. If we differentiated $f'(x)$ again (finding the second derivative, $f''(x)$ ), we would get the acceleration. These concepts are the backbone of understanding motion and dynamics. Even in economics, derivatives are used to model marginal utility, marginal revenue, and marginal cost, helping businesses make informed decisions about pricing and production. The ability to calculate and interpret derivatives unlocks a deeper understanding of how systems change and evolve.

A Quick Recap and Further Exploration

We've successfully found the derivative of $f(x)= rac{x^2-2}{x-1}$ using the Quotient Rule. Remember the key steps: identify your numerator and denominator, find their individual derivatives, and then apply the Quotient Rule formula: $rac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$ . Simplifying the resulting expression is also a crucial part of the process. Practice is key to mastering this. Try differentiating other rational functions, and don't be afraid to revisit the power rule and the Quotient Rule until they feel comfortable.

Calculus is a vast and fascinating subject. If you found this exercise interesting, you might want to explore other differentiation rules, such as the product rule and the chain rule, which are essential for differentiating more complex functions. Understanding the geometric interpretation of the derivative as the slope of the tangent line can also greatly enhance your intuition. For those interested in the broader applications of calculus, exploring topics like integration, differential equations, and multivariable calculus can open up even more exciting avenues of study.

For more in-depth information on derivatives and calculus, I recommend visiting Khan Academy or Paul's Online Math Notes. These resources offer comprehensive explanations, examples, and practice problems that can further solidify your understanding.