Problem 44 - Calculus and its applications

Question

Find \(f^{\prime}(x)\) for the given function. \(f(x)=\frac{1}{x^{2}}\) (See Exercise 52 in Section 1.3.)

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{-2}{x^3} \).

Step 1: Identify the function and rewrite it in derivative-friendly form

The given function is \( f(x) = \frac{1}{x^2} \). To differentiate it more easily, rewrite it as \( f(x) = x^{-2} \). This form will allow us to apply the power rule directly.

Step 2: Apply the power rule for differentiation

The power rule states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \). Here, \( n = -2 \), so apply the rule: \( f'(x) = -2 \cdot x^{-2-1} = -2 \cdot x^{-3} \).

Step 3: Simplify the result

The expression \( -2 \cdot x^{-3} \) can be rewritten as \( \frac{-2}{x^3} \). This is the simplified form of the derivative \( f'(x) \).

Key Concepts

Power Rule for Differentiation

Understanding the power rule is essential for differentiating functions efficiently. Whenever you have a function in the form of \( x^n \), where \( n \) is any real number, the power rule can be applied to find its derivative. The rule itself is quite simple: if \( f(x) = x^n \), then the derivative, \( f'(x) \), is given by \( n \cdot x^{n-1} \). This means that you bring down the power as a coefficient and then subtract one from the power to find the derivative.

In the original exercise, the function provided was \( f(x) = \frac{1}{x^2} \). To make it derivative-friendly, we rewrote it in the form \( f(x) = x^{-2} \) before applying the power rule. Doing so allowed us to easily identify \( n = -2 \) and apply the rule directly, giving us \( f'(x) = -2 \cdot x^{-3} \).

This is a straightforward application of the power rule that shows how it simplifies the differentiation process for functions expressed as powers of \( x \).

Derivative of a Function

The derivative of a function measures how that function changes as its input changes. It provides the slope or gradient of the function at any given point. Knowing how to find the derivative is critical for understanding the behavior of functions.

For a function like \( f(x) = \frac{1}{x^2} \), finding the derivative involves first rewriting it to a more convenient form as \( f(x) = x^{-2} \). Then, by applying the power rule, you find \( f'(x) = -2 \cdot x^{-3} \). This derivative tells us how \( f(x) \) changes with respect to \( x \).

Derivatives are significant due to several reasons:

• They allow us to find rates of change.
• They help determine the function’s increasing or decreasing behavior.
• They are used in finding local maxima and minima, aiding in sketching graphs of functions.

Derivatives are a fundamental concept in calculus and many applications, such as physics, engineering, and economics.

Simplifying Derivatives

Simplifying derivatives is an important step in ensuring that the derivative is in its most understandable form. Once you have found the derivative using rules like the power rule, you should always look to simplify the expression for clarity and usability.

In the example \( f'(x) = -2 \cdot x^{-3} \), simplifying involves changing it from an exponential form to a fraction. This is done by recognizing that \( x^{-3} \) can be rewritten as \( \frac{1}{x^3} \). Thus, we simplify \( -2 \cdot x^{-3} \) to \( \frac{-2}{x^3} \).

Simplifying derivatives makes them easier to interpret and apply, especially when solving problems such as finding the tangent line at a point or integrating the function later on. It also enhances readability and aids in subsequent mathematical operations.