Problem 20 - Precalculus: Mathematical for Calculus

Question

Find \(f^{\prime}(a),\) where \(a\) is in the domain of \(f .\) $$f(x)=-\frac{1}{x^{2}}$$

Step-by-Step Solution

Answer

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

Step 1: Identify and Understand the Derivative Formula

The derivative of a function \(f(x)\) with respect to \(x\) is found using the formula \(f^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}\). In this case, we need to find \(f^{\prime}(a)\), so we'll first find \(f^{\prime}(x)\).

Step 2: Rewrite the Function

Given the function \(f(x) = -\frac{1}{x^2}\), it can be rewritten as \(f(x) = -x^{-2}\). This form will be helpful when applying the power rule for derivatives.

Step 3: Apply the Power Rule

Use the power rule for differentiation, which states that \(\frac{d}{dx}[x^n] = nx^{n-1}\). Applying this to \(-x^{-2}\), we get: \[f'(x) = -(-2)x^{-2-1} = 2x^{-3}\].

Step 4: Substitute \(a\) into the Derivative

We have found \(f'(x)\), now substitute \(a\) into this expression: \[f'(a) = 2a^{-3}\].

Step 5: Write the Final Answer

Thus, the derivative of \(f(x) = -\frac{1}{x^2}\) with respect to \(x\), evaluated at \(a\) is: \[f^{\prime}(a) = \frac{2}{a^{3}}\].

Key Concepts

Power Rule

The power rule is one of the most fundamental tools in calculus, particularly when working with derivatives. It's a quick method to find the derivative of power functions. If you have a function of the form \( f(x) = x^n \), the derivative \( f'(x) \) is found by multiplying the exponent \( n \) by \( x \) to the power of \( n-1 \). This can be summarized as:

• \( \frac{d}{dx}[x^n] = nx^{n-1} \)

Applying the power rule significantly simplifies the process of differentiation.
Consider the function in our exercise: \( f(x) = -\frac{1}{x^2} \). By rewriting it as \( f(x) = -x^{-2} \), we can easily apply the power rule. Multiply the exponent, \(-2\), by the starting function \(-x\), and reduce the exponent by one unit:

• \( f'(x) = -(-2)x^{-3} \)

The rule efficiently gives us the result \( f'(x) = 2x^{-3} \), demonstrating its power and effectiveness.

Function Differentiation

Differentiation is the process of finding a function's derivative, which measures how the function's output changes as the input changes. In simple terms, it tells you how fast the function is changing at any given point.
When differentiating, especially complex expressions, it's helpful to break down the function into simpler parts. For example, write \( f(x) = -\frac{1}{x^2} \) as \( f(x) = -x^{-2} \). Such transformation allows for easier application of differentiation rules like the power rule.
This approach provides clarity and simplicity, enabling you to handle more challenging functions. Awareness of such techniques can greatly enhance your understanding and efficiency in calculus.

Limit Definition of Derivative

The limit definition of a derivative provides the theoretical underpinning for differentiation. It is defined as:

• \( f^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \)

This definition shows how the average rate of change of a function \( f \) over the interval \([x, x+h]\) approaches the instantaneous rate of change as \( h \to 0 \).
In practice, while using the limit definition directly can be cumbersome, it ensures the precision and validity of derivative calculations.
In our exercise, identifying that we're essentially looking for an expression \( f^{\prime}(x) \) first, before substituting any value like \( a \), is crucial. It verifies our process, ensuring that the steps prior to applying rules like the power rule adhere to the foundational calculus concepts.
Thus, even though more straightforward methods like the power rule are typically used, the limit definition remains a cornerstone of understanding derivatives.