Question

Find the slope of the tangent or the rate of change at the given value of \(x\) $$y=x+\frac{1}{x} \text { at } x=2$$

Short answer

The slope of the tangent at \(x=2\) is \(\frac{3}{4}\).

Step-by-step solution

Identify the Function

We are given the function \(y = x + \frac{1}{x}\) and need to find the rate of change or the slope of the tangent at the point where \(x = 2\).

Differentiate the Function

To find the slope of the tangent, we need to differentiate the given function with respect to \(x\). The derivative of \(y = x + \frac{1}{x}\) using the power rule and the derivative of a constant multiplied by a function is: \(\frac{dy}{dx} = \frac{d}{dx}(x) + \frac{d}{dx}(\frac{1}{x}) = 1 - \frac{1}{x^2}\).

Evaluate the Derivative at \(x=2\)

Next, we substitute \(x = 2\) into the derivative \(\frac{dy}{dx} = 1 - \frac{1}{x^2}\) to find the slope of the tangent at this point: \(\frac{dy}{dx} \bigg|_{x=2} = 1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4}\).

Key concepts

Differentiate the Function

Differentiating a function essentially means finding the derivative of that function with respect to a variable - in this case, with respect to 'x'. The derivative represents the rate at which the function's value is changing at any given point. In simpler terms, when you differentiate a function, you're calculating how much 'y' will change for a small change in 'x'.

For the function provided in the exercise, \(y = x + \frac{1}{x}\), differentiation gives you a formula that can tell you the slope of the tangent line to the curve represented by the function at any point on it. Remember this fundamental concept: a derivative is like a snapshot of a function's behavior at a specific instant. It provides instantaneous rate of change, and indeed, calculating it is a cornerstone of calculus.

Rate of Change

The 'rate of change' is a critical concept in both mathematics and real-world applications. It describes how one quantity changes in relation to another. When you're looking at a function, the rate of change equates to the slope of the tangent line at any given point on the function's graph. This slope tells you how steep or flat the function is at that point. In practical scenarios, it could represent speed, acceleration, or the growth rate of a population, among endless other possibilities.

To find the rate of change for our function \(y = x + \frac{1}{x}\) at \(x = 2\), we used the function's derivative. By plugging \(x = 2\) into the derivative, we obtain the specific rate of change at that point, giving us the slope of the tangent \(\frac{3}{4}\), which is critical for understanding not just the function's behavior at \(x = 2\), but also the exact steepness of the curve of the graph at that particular spot.

Power Rule of Differentiation

The power rule is a basic yet powerful tool in calculus used to differentiate functions that are polynomials or have terms that can be represented as \(x^n\), where 'n' is any real number. In essence, it states that the derivative of \(x^n\) is \(nx^{n-1}\). For instance, applying the power rule to \(x^3\) would give us \(3x^2\).

When we differentiated the function \(y = x + \frac{1}{x}\), we used the power rule to find the derivative. The term \(x\) is a power function with 'n' equals 1, so its derivative is simply 1 (since \(x^0 = 1\)). For the term \(\frac{1}{x}\), which is the same as \(x^{-1}\), applying the power rule gives us a derivative of \(\frac{-1}{x^2}\). This approach simplifies the process of differentiation and allows us to easily find the slope of the tangent line at any point on the graph of a power function.