Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ D_{t}\left((F(t))^{-2}\right) $$

Step-by-Step Solution

Answer

\(-2(F(t))^{-3} \cdot F'(t)\).

Step 1 - Understand the Problem

We need to find the derivative of the function \((F(t))^{-2}\) with respect to \(t\). We know \(F\) is a differentiable function.

Step 2 - Apply the Chain Rule

To differentiate \((F(t))^{-2}\), use the chain rule. The chain rule states that \(\frac{d}{dt}[u(t)^n] = n \cdot u(t)^{(n-1)} \cdot \frac{du}{dt}\), where \(u(t) = F(t)\) and \(n = -2\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that focuses on understanding how functions change. When we differentiate a function, we're finding the rate at which it changes, which is known as the derivative. Differentiation is immensely useful in physics, economics, and many other fields to model and predict real-world phenomena.

To differentiate a function, we apply specific rules and formulas. For instance, the power rule helps when dealing with polynomial expressions, where if you have a term like \(x^n\), its derivative will be \(nx^{n-1}\). However, not all functions fit into simple forms, so we sometimes need more advanced techniques, such as the chain rule, which we'll discuss in detail later.

For a function \(F(t)\), its derivative with respect to \(t\), denoted as \(D_t[F(t)]\) or \(\frac{dF}{dt}\), tells us how \(F(t)\) changes as \(t\) changes. Every time we perform differentiation, the goal is to simplify this process and understand the behavior of the function over its domain.

Chain Rule

The chain rule is an essential tool in differentiation, especially when dealing with composite functions. It helps us break down complex functions into simpler parts. In simple terms, if you have a function that consists of another function inside it, the chain rule enables you to find the derivative efficiently.

Consider the function \(y = (F(t))^{-2}\), where \(F(t)\) is a differentiable function. To apply the chain rule here, we recognize \((F(t))^{-2}\) as a composition of functions \(u(t)^n\), where \(u(t) = F(t)\) and \(n = -2\). The chain rule formula is:

• \(\frac{dy}{dt} = n \, ext{(the power)} \cdot u(t)^{n-1} \cdot \frac{du}{dt}\)

By applying this, we differentiate the outer function \(u(t)^n\), adjust the power \(n\), and then multiply by the derivative of the inner function \(u(t) = F(t)\). This method efficiently handles complex derivatives and is a critical technique for those studying calculus.

Function Derivatives

Function derivatives give us insights into how functions behave and change. When a function like \(F(t)\) is given, its derivative \(\frac{dF}{dt}\) informs us about the rate of change of the function's output with respect to the input \(t\).

Understanding function derivatives is crucial for solving many real-world problems. For example, in physics, derivatives can describe the speed of an object over time. In our exercise, finding the derivative of \((F(t))^{-2}\) involves using the chain rule to handle this complex function, as the derivative of \((F(t))^{-2}\) is not straightforward.

Using differentiation, and particularly with the help of rules like the chain rule, we can express these changes mathematically and predict future behavior. This process not only aids in academic exercises but also supports practical applications in science, engineering, and beyond.