Find the slope of the following ct \( y=\frac{1}{x-2} \) The slope of the given curve at \( x \) (Simplify your answer.)

To find the slope of the curve \( y = \frac{1}{x - 2} \), we first need to find its derivative. Using the power and chain rule, we can differentiate: \[ \frac{dy}{dx} = \frac{d}{dx}(x - 2)^{-1} = -1(x - 2)^{-2} \cdot (1) = -\frac{1}{(x - 2)^2} \] So, the slope of the given curve at any point \( x \) is: \[ -\frac{1}{(x - 2)^2} \] This means that as \( x \) approaches 2, the slope becomes steeper (or more negative) since you're dividing by a smaller number! Just watch out for that vertical asymptote! Now, let's explore the history of calculus. Did you know that the concept of derivatives wasn't standardized until the late 17th century? Mathematicians like Newton and Leibniz independently developed these ideas, leading to the powerful tools we use today! For real-world application, the formula for the slope can help you analyze things like how steep a hill is or the rate of change in stock prices. Knowing how to find the slope can guide you in making important decisions, whether you're taking a road trip or managing investments!