Question

In Exercises \(5-8,\) use the definition \(f^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\) to find the derivative of the given function at the indicated point. $$f(x)=1 / x, a=2$$

Short answer

The derivative of \(f(x) = 1/x\) at \(a = 2\) is \(-1/4\).

Step-by-step solution

Write Down the Definition of the Derivative

The definition of the derivative at a point \(a\) is given by \[f^{\prime}(a) = \lim_{{x \rightarrow a}} \frac{f(x) - f(a)}{x - a}\]This is the formula we are going to use to find the derivative of our function \(f(x) = 1/x\).

Substitute the Function Into the Definition

Now we substitute our function \(f(x) = 1/x\) and the given point \(a = 2\) into the formula. This gives us: \[\lim_{{x \rightarrow 2}} \frac{1/x - 1/2}{x - 2}\]

Simplify the Fraction Inside the Limit

Now we can simplify the expression inside the limit by finding a common denominator for the terms in the numerator:\[\lim_{{x \rightarrow 2}} \frac{2 - x}{2x(x - 2)}\]

Decrease the Degree of the Numerator and Denominator

We see that \(2 - x\) in the numerator and \(x - 2\) in the denominator are negatives of each other. If we factor out -1 from either of them, they can cancel out. Do this to get:\[\lim_{{x \rightarrow 2}} \frac{-1}{2x}\]

Calculate the Limit

With the simplified form, we can now substitute \(x = 2\) into the formula: \[f^{\prime}(2) = -1/(2*2) = -1/4\]

Key concepts

Definition of the Derivative

Understanding the derivative is key to mastering calculus. Simply put, the derivative measures how a function's output value changes as its input value changes. Imagine being in a car and watching the speedometer; the derivative would represent the speed, showing how quickly your position changes over time.

The mathematical definition of a derivative at a particular point, written as \(f'(a)\), is the limit of the average rate of change of the function as the interval over which we measure this rate shrinks to zero. It is formally defined by the expression:
\[f'(a) = \lim_{{x \rightarrow a}} \frac{f(x) - f(a)}{x - a}\]
This equation tells us that to find the instantaneous rate of change of \(f(x)\) at some point \(a\), we examine the ratio of the change in \(f(x)\) to the change in \(x\) as \(x\) gets infinitely close to \(a\).

When you encounter a problem, like finding the derivative of \(f(x) = 1/x\) at \(x = 2\), the definition is your starting point. You'll replacethe function and point into the expression and follow through the process of taking the limit, which ultimately reveals the derivative at that point.

Limit of a Function

The concept of a limit is foundational in calculus. It describes the behavior of a function as it approaches a certain point, but isn't necessarily defined at that point. It answers the question: 'What value does the function get close to as the input approaches a certain number?'

For instance, when calculating the derivative of \(f(x)=1/x\) at \(a=2\), you calculate the limit as \(x\) approaches 2. This process helps us understand the behavior of the function near \(x = 2\) without actually letting \(x\) equal 2, especially since direct substitution can sometimes lead to undefined expressions.

In our derivative calculation problem, you might recall seeing this step:
\[\lim_{{x \rightarrow 2}} \frac{1/x - 1/2}{x - 2}\]
This is where we apply the concept of the limit to evaluate the derivative. The limit tells us the value that the fraction is approaching as \(x\) becomes very close to 2, which is essential for understanding the derivative at that specific point.

Simplifying Mathematical Expressions

When working with algebra and calculus problems, simplification allows us to make expressions easier to understand and solve. A simpler form can also reveal patterns or properties that aren't immediately visible in the original format.

In the context of calculating derivatives using limits, simplification is crucial for getting to a point where the limit can actually be computed. For example, when you simplify the expression \(\frac{1/x - 1/2}{x - 2}\), you're looking to eliminate complex fractions, cancel common factors, and reduce the expression to something that can be evaluated directly. Here's how you might do it step by step:

• Find common denominators to combine fractions: \(\frac{2 - x}{2x(x - 2)}\).
• Recognize that \(2 - x\) and \(x - 2\) are negatives of one another; factor out -1 to cancel them.
• Arrive at a much simpler expression: \(\frac{-1}{2x}\).

Following these steps not only simplifies the calculation of the limit but also clearly demonstrates the underlying mathematical concepts at play.