Problem 46 - A Graphical Approach to Precalculus with Limits

Question

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=x^{2}-5 x$$

Step-by-Step Solution

Answer

The formula for \(f^{\prime}(a)\) is \(2a - 5\).

Step 1: Understand the Problem

The problem requires us to find the derivative, represented as \(f^{\prime}(a)\), of the function \(f(x) = x^2 - 5x\) at a general point \(x = a\). This involves applying the rules of differentiation to find \(f^{\prime}(x)\) and then substituting \(x = a\) into this derivative.

Step 2: Differentiate the Function

We begin by differentiating the given function \(f(x) = x^2 - 5x\). Using the power rule, the derivative of \(x^2\) is \(2x\), and the derivative of \(-5x\) is \(-5\). Therefore, \[f^{\prime}(x) = 2x - 5.\]

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

Now that we have the derivative \(f^{\prime}(x) = 2x - 5\), we substitute \(x = a\) into the derivative to find \(f^{\prime}(a)\). So \[f^{\prime}(a) = 2a - 5.\]

Key Concepts

Understanding the Power Rule

The power rule is a fundamental principle in differentiation used to find the derivative of a function with a power term. When you need to differentiate a term like \(x^n\), the power rule simplifies the process.
Just multiply the entire term by the power, and then subtract one from the power.

• If you have the function \(f(x) = x^n\), using the power rule gives you the derivative as \(f'(x) = nx^{n-1}\).
• This means you bring the power down as a coefficient and reduce the power by one.

In the given exercise, the function \(f(x) = x^2 - 5x\) consists of two terms. Using the power rule, we find that:

• The derivative of \(x^2\) is \(2x\).
• The derivative of \(-5x\) is simply \(-5\) since \(-5x\) is a linear term (can be taken as \(x^1\)).

Thus, by applying the power rule to each term separately, you quickly reach the function's derivative.

Concept of Function Derivative

The derivative of a function gives you a new function that represents the rate of change, or the slope, of the original function at any given point. This concept is crucial in calculus as it highlights how a small change in the input affects the output.
In the mathematical sense, if you start with a function \(f(x)\), the derivative \(f'(x)\) shows the instant rate of change at each point.

• It helps in understanding how the function behaves at any specific interval of \(x\).
• The derivatives allow us to find the slope of the tangent line at any point on the function curve.

For the problem \(f(x) = x^2 - 5x\), its derivative \(f'(x) = 2x - 5\) can be used to understand how the original function changes as x moves.

Evaluating the Derivative

Once you've found the derivative of a function, you could be asked to evaluate it at a specific point. Evaluating a derivative involves substituting a particular value of \(x\) into the derivative function.
This provides you the rate of change of the function at that particular point, essentially detailing how steep or flat the curve is at that spot.

• For example, after finding the derivative \(f'(x) = 2x - 5\), you might need \(f'(a)\).
• This just means you replace \(x\) with \(a\) in the derivative, obtaining \(f'(a) = 2a - 5\).

In our exercise, this gives a clear, numeric answer for the rate of change at \(x = a\). Evaluating the derivative completes the process, providing insight into the behavior of the original function at particular points.