Problem 57 - Calculus an Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ g(t)=\frac{1}{t^{2}-2} $$

Step-by-Step Solution

Answer

The derivative of the function \(g(t) = \frac{1}{t^{2}-2}\) is \(g'(t) = -\frac{2t}{(t^{2}-2)^{2}}\). The derivative was found using the Chain Rule and the Power Rule.

Step 1: Rewrite the function

First, rewrite the function as \(g(t) = (t^{2}-2)^{-1}\) to simplify it for differentiation.

Step 2: Identify the outer and inner functions

In order to apply the Chain Rule, identify the outer function as \(f(x) = x^{-1}\) and the inner function as \(u(t) = t^{2}-2\).

Step 3: Apply the Chain Rule

Differentiate the outer function with respect to \(u(t)\), then multiply by the derivative of \(u(t)\). Using the chain rule gives: \(g'(t) = f'(u(t)) \cdot u'(t) = -1 \cdot (t^{2}-2)^{-2} \cdot (2t)\).

Step 4: Simplify the derivative

Now, simplify the derivative to get \(g'(t) = -\frac{2t}{(t^{2}-2)^{2}}\).

Key Concepts

Derivative of a Function

Differentiation is a fundamental concept in calculus, representing the rate at which a function is changing at any given point. When we talk about the derivative of a function, we refer to this concept of change. For a function, say, \( y = f(x) \), its derivative is usually denoted by \( f'(x) \) or \( \frac{dy}{dx} \). This derivative tells us how the function's output value changes with respect to a change in the input value.

In the process of finding the derivative of a function, one typically encounters various scenarios that require different rules or techniques to compute this rate of change successfully. Concerning the exercise given, the function \( g(t) \) is a composite function, which means it is a function within another function. For such functions, simple differentiation rules would not suffice, and specific methods - such as the Chain Rule - are necessary to find the derivative correctly.

Chain Rule

The Chain Rule is a powerful differentiation rule that allows us to find the derivative of composite functions. It essentially tells us how to differentiate a function of a function. To apply the Chain Rule, one must identify the outer function (the function that contains another function) and the inner function (the function inside the outer function).

Let's denote the outer function as \( f(u) \) and the inner function as \( u(x) \). According to the Chain Rule, the derivative of the composite function \( f(u(x)) \) is found by taking the derivative of the outer function with respect to the inner function \( u \), and then multiplying it by the derivative of the inner function with respect to \( x \). In mathematical terms, this is written as:

\( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \)

The exercise above demonstrates the Chain Rule by identifying the function \( (t^{2}-2)^{-1} \) as the outer function and \( t^{2}-2 \) as the inner function. The derivative is then computed using this approach, allowing for the rate of change of the composite function to be accurately determined.

Power Rule

Another important rule for differentiation is the Power Rule, which makes finding the derivatives of polynomial functions significantly simpler. The Power Rule states that if you have a function \( f(x) = x^n \), where \( n \) is any real number, the derivative of that function is \( f'(x) = nx^{n-1} \). This means you multiply the exponent \( n \) by the base function and then decrease the exponent by one.

For instance, if you're differentiating \( u(t) = t^2 \) as part of the exercise, applying the Power Rule gives us the derivative \( u'(t) = 2t \). The Power Rule is thus an essential tool in calculus, allowing us to quickly find the derivatives of functions with terms raised to a power, and in the provided exercise, it simplifies the process of differentiating the inner function of our composite function.