Question

Find the slope of the tangent or the rate of change at the given value of \(x\) $$y=\frac{1}{x} \text { at } x=3$$

Short answer

The slope of the tangent to the function at x=3 is $$- \frac{1}{9}.$$

Step-by-step solution

Determine the function to differentiate

Identify the function for which the slope of the tangent is to be found. In this case, the function given is: $$y = \frac{1}{x}.$$

Differentiate the function with respect to x

Use the power rule to differentiate the function with respect to x. The power rule states that the derivative of $$x^n$$ is $$nx^{n-1}.$$ Since $$y = x^{-1},$$ the derivative is: $$y' = -1 \times x^{-2} = -\frac{1}{x^2}.$$

Evaluate the derivative at the given value of x

Plug the given value of x into the derivative to find the slope of the tangent at that point. For $$x=3,$$ the slope is: $$m = -\frac{1}{3^2} = -\frac{1}{9}.$$

Key concepts

Differentiation

At its core, differentiation is a mathematical process that finds how a function changes at any point. For students tackling calculus, understanding differentiation is crucial because it gives insight into the rate at which things change. For instance, if you're looking at how quickly a car accelerates, differentiation can tell you the car's speed at any given moment. This concept is often visually represented on a graph by the slope of the tangent line to a curve at any specific point.

In the classroom, when you're given a smooth curve that represents a function, such as a line on a graph showing the growth of a plant over time, differentiation enables you to predict the plant's growth rate at any instant. The derivative is the function that results from the differentiation process and can be thought of as a formula that gives the slope of the tangent line at any point along the original curve. This slope is what we refer to as the 'rate of change' at that particular point.

Power Rule for Derivatives

The power rule is a shortcut that simplifies the process of finding the derivative of power functions, where 'power functions' are functions with a single term like \(x^n\). Usually, in calculus, we use the power rule because it's much easier and quicker than calculating derivatives from first principles.

To apply the power rule, you simply take the exponent (\(n\)) of the \(x^n\) term, multiply the whole term by \(n\), and then subtract one from the exponent. In mathematical terms, the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\). With this rule in mind, solving problems like finding the slope of a tangent becomes a systematic and straightforward task. It's a powerful tool aiding students to navigate calculus problems involving rates of change efficiently.

Rate of Change

In everyday terms, the 'rate of change' refers to how quickly something changes over time or another variable. In mathematics, and more specifically in calculus, this concept is represented as the derivative of a function. It provides important real-world applications, such as determining speed, acceleration, or growth rates.

When solving problems in calculus, the rate of change is visually represented by the steepness or slant of a tangent line to a curve at a point. For instance, if you were to graph the path of a basketball thrown into the air, the rate of change at any point on the graph would tell you the basketball's vertical velocity at that particular instant. Understanding how to find and interpret the rate of change is essential for students to solve problems across natural and social sciences, as well as in economics and engineering.