Problem 22 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use the limit de nition to differentiate \(f(x)=\frac{1}{x^{2}}\).

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=\frac{1}{x^{2}}\) using the limit definition is \(f'(x) = -2x^{-3}\) or equivalently \(f'(x) = -\frac{2}{x^{3}}\).

Step 1: Rewrite the function

It is easier to differentiate the function if we rewrite \(f(x) = \frac{1}{x^{2}}\) as \(f(x) = x^{-2}\).

Step 2: Apply the definition of the derivative

The limit definition of the derivative is given by \[f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}\]. Let's substitute the function \(f(x)\) and simplify the limit.

Step 3: Calculate \(f(x + h)\)

First, we need to calculate the function \(f(x + h) = (x + h)^{-2}\).

Step 4: Simplify numerator of the limit

The numerator of the limit is \(f(x + h) - f(x) = (x + h)^{-2} - x^{-2}\). To simplify, find a common denominator and simplify the fraction.

Step 5: Simplify the limit

Plug in the simplified numerator into the limit definition and simplify the limit. To further simplify, separate the terms and use the fact that the limit of a sum equals the sum of the limits.

Step 6: Evaluate the limit

Now evaluate the limit as h approaches 0.

Key Concepts

Differentiate Functions

Differentiation is a fundamental process in calculus which measures how a function changes as its input changes. To differentiate a function means to find its derivative, which is a formula that provides the slope of the function at any given point. The derivative of a function at a particular point represents the rate at which the function's value is changing at that point.

In the context of the exercise given, we're asked to differentiate the function f(x) = 1/x², which can also be written as f(x) = x^-2. This form is more conducive to differentiation as it directly fits into the power rule for derivatives. However, we're asked to use the limit definition, which involves calculating the limit of a difference quotient as h, an increment to x, approaches zero. This definition requires the understanding of limits, which is another core concept in calculus. Understanding how to evaluate these limits is essential for successfully differentiating functions using this definition.

Simplifying Expressions

When working with mathematical expressions, especially in the context of differentiation and calculus, it's often necessary to simplify expressions to make the problem more manageable. Simplification may involve expanding polynomials, combining like terms, or finding common denominators for fractions.

In the example exercise, simplification is a critical step that helps in evaluating the limit definition of the derivative. When the function f(x + h) = (x + h)^-2 is subtracted from f(x) = x^-2, we get the difference quotient. This expression can appear complex, but by finding a common denominator, the expression is simplified, enabling us to proceed to the limit evaluation. Skilled manipulation and simplification of expressions are essential in calculus; they help us prepare an expression so that we can apply further mathematical operations, such as taking limits.

Evaluating Limits

The concept of evaluating limits is central to understanding the behavior of functions as the input approaches a particular value. Limits are foundational in calculus and are used in defining derivatives and integrals.

In the limit definition of the derivative, you calculate the limit of a difference quotient as h tends to zero. To evaluate this limit, you may need to apply techniques such as factoring, using common denominators, or l'Hôpital's Rule when faced with indeterminate forms. In the step-by-step example, after simplifying the expressions, you can often just substitute the value of h as zero to find the limit. But sometimes, you may encounter forms like 0/0, which require further simplification before substitution. The ability to evaluate these limits properly is crucial for finding the derivative of a function using the limit definition.