Problem 34 - Calculus: Early Transcendentals

Question

Find \( f'(a) \). \( f(x) = x^{-2} \)

Step-by-Step Solution

Answer

\( f'(a) = -2a^{-3} \)

Step 1: Recall the Power Rule for Differentiation

The power rule for differentiation states that if you have a function of the form \( f(x) = x^n \), where \( n \) is a constant, then its derivative is \( f'(x) = n \cdot x^{n-1} \). In this case, \( f(x) = x^{-2} \), with \( n = -2 \).

Step 2: Apply the Power Rule

Using the power rule on \( f(x) = x^{-2} \), we differentiate it to get \( f'(x) = -2 \cdot x^{-2-1} \). This simplifies to \( f'(x) = -2 \cdot x^{-3} \).

Step 3: Evaluate the Derivative at \( x = a \)

To find \( f'(a) \), substitute \( a \) into the derivative. This gives \( f'(a) = -2 \cdot a^{-3} \).

Key Concepts

Understanding the Power Rule

The power rule is a basic yet powerful tool used in differentiation. This rule makes it easy to find the derivative of functions that have variables raised to any real number power. Simply put, if you have a function expressed as \( f(x) = x^n \), where \( n \) is a constant, you can find the derivative using the power rule formula: \( f'(x) = n \cdot x^{n-1} \).
For example, consider the function \( f(x) = x^{-2} \). Here, you apply the power rule by:

• Identifying \( n = -2 \), which is the power of \( x \).
• Calculating the derivative as \( f'(x) = -2 \cdot x^{-2-1} \).
• Simplifying to get \( f'(x) = -2 \cdot x^{-3} \).

By following these steps, applying the power rule becomes straightforward, allowing you to quickly find derivatives of functions with power terms.

Derivative and Its Significance

A derivative helps us understand how a function behaves as its input changes. It tells us the rate of change or the slope of the function at any given point. This concept is widely used in various fields like physics, engineering, and even economics.
In the example exercise, the function \( f(x) = x^{-2} \) is differentiated to find \( f'(x) \). Applying the power rule results in \( f'(x) = -2 \cdot x^{-3} \). This derivative expression shows:

• The function is decreasing where the derivative is negative.
• The steepness or rate of decrease is determined by \( -2 \cdot x^{-3} \).
• The negative sign indicates a downward slope.

Understanding derivatives enables us to figure out how functions increase or decrease and helps in finding critical points and inflection points.

Function Evaluation at a Specific Point

To evaluate a derivative at a specific point, you substitute the value into the derivative function. This process tells you the rate of change of the original function at the chosen point.

• In the example, to find \( f'(a) \), you substitute \( a \) into \( f'(x) = -2 \cdot x^{-3} \).
• This gives \( f'(a) = -2 \cdot a^{-3} \).
• This is known as evaluating the function at \( x = a \).

Evaluating the derivative at \( x = a \) provides insights about the function at that particular point. For example, it can show whether the function is increasing or decreasing at point \( a \) and how rapidly this change is occurring.