Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(t)=\frac{t^{2}-1}{t+4} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(t)=\frac{t^{2}-1}{t+4}\) is \(f'(t) = \frac{t^{2}+8t + 1}{(t+4)^{2}}\). The rule of differentiation used here was the Quotient Rule.

Step 1: Identification

The function given is \(f(t)=\frac{t^{2}-1}{t+4}\). This function is in the form of a quotient, hence the Quotient Rule of differentiation will be applied.

Step 2: Application of the Quotient Rule

\(f'(t) = \frac{(2t)(t+4) - (1)(t^{2}-1)}{(t+4)^{2}} = \frac{2t^{2}+8t - t^{2} + 1}{(t+4)^{2}} = \frac{t^{2}+8t + 1}{(t+4)^{2}}\). Here, the derivative of \(t^{2}-1\) is \(2t\) and the derivative of \(t+4\) is \(1\).

Step 3: Simplify the derivative

Upon simplifying, the derivative \(f'(t)\) simplifies to \(\frac{t^{2}+8t + 1}{(t+4)^{2}}\).

Key Concepts

Derivative of a Function

When we talk about the derivative of a function, we're looking at the rate at which it changes at any given point. It's a fundamental concept in calculus, representing the slope of the tangent line to the function's graph at a particular point. Conceptually, think of it as the instant speedometer reading if your function was the path of a car: how fast are we going right this second?

In mathematical terms, the derivative of a function at a point is the limit of the average rate of change as the interval over which we measure shrinks to zero. A derivative tells you how a function is behaving in terms of increasing or decreasing, and by how much. If we have a function like our example, \(f(t) = \frac{t^2-1}{t+4}\), we're interested in finding \(f'(t)\), the derivative of the function with respect to \(t\).

Applying the Quotient Rule

The Quotient Rule is a method for finding the derivative of a function that's expressed as the quotient of two other functions. The rule is an invaluable shortcut when faced with dividing functions, especially compared to using the limit definition of the derivative.

Here's the rule in a general form:

Quotient Rule Formula

For two functions \(u(t)\) and \(v(t)\), the derivative of the quotient \(\frac{u(t)}{v(t)}\) is:\[\frac{d}{dt}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^{2}}\]Where \(u'\) is the derivative of \(u\) with respect to \(t\), and \(v'\) is the derivative of \(v\) with respect to \(t\). In our example, we apply this rule by identifying \(u(t) = t^2 - 1\) and \(v(t) = t + 4\). Taking derivatives separately for \(u(t)\) and \(v(t)\) and then applying the quotient rule gives us the derivative of the original function.

Simplifying the Derivative

After applying the quotient rule, the next step is simplifying the derivative to make it easier to understand and work with. Simplification may involve combining like terms, factoring, or expanding expressions. The aim is to present the derivative in its simplest, most digestible form. By simplifying, we avoid mistakes in further calculations and make it easier to evaluate the derivative at particular points.

In the case of our example, the initial application of the Quotient Rule resulted in \(\frac{2t^2+8t-t^2+1}{(t+4)^{2}}\), which includes like terms in the numerator that can be combined. Upon simplification, the derivative becomes \(\frac{t^2+8t+1}{(t+4)^2}\). This final expression is much easier to evaluate for particular values of \(t\), analyse, and use in subsequent problems.