Problem 29 - Calculus an Applied Approach

Question

Find the value of the derivative of the function at the given point. $$ {f(x)=\frac{1}{x}}\quad {(1,1)} $$

Step-by-Step Solution

Answer

The value of the derivative of the function at the point (1,1) is -1.

Step 1: Calculate Derivative

To calculate the derivative of \(\frac{1}{x}\), remember that this can be written as \(x^{-1}\). Apply the power rule for differentiation which states that the derivative of \(x^n\) is \(nx^{n - 1}\). So the derivative \(f'(x)\) is \(-x^{-2}\) or \(-\frac{1}{x^2}\).

Step 2: Evaluate Derivative at the Given Point

The point is given as (1,1). In this case, the x-coordinate is 1. So, substitute \(x = 1\) into the derivative. \(f'(1) = -\frac{1}{1^2} = -1\).

Key Concepts

Power Rule

Differentiating functions is a major component in calculus, and the power rule is one of the simplest and most versatile tools for this. The power rule helps to quickly find the derivative of functions where the variable is raised to any power. If you have a function like \( f(x) = x^n \), the power rule states that the derivative of this function, \( f'(x) \), is \( nx^{n-1} \).

Let's see how this applies to our example. The given function is \( f(x) = \frac{1}{x} \). By rewriting it as \( x^{-1} \), we can easily apply the power rule:

• The original power, \( n \), is -1.
• The derivative of \( x^{-1} \) is calculated by taking \( -1 \times x^{-2} \).
• This simplifies to \( f'(x) = -x^{-2} \) or equivalently \( -\frac{1}{x^2} \).

Recognizing and appropriately rewriting the function to fit the power rule can simplify many differentiation problems!

Function Evaluation

Once you have the derivative of a function, the next step is often to evaluate it at a specific point. This helps us understand the function's behavior at that specific location, such as the slope of the tangent line.

In our problem, we need to find the derivative at the point \((1,1)\). The x-coordinate is what we're interested in when evaluating the derivative, which is \( x = 1 \). By substituting this value into the derivative \( f'(x) = -\frac{1}{x^2} \), we get:

• Substitute \( x = 1 \).
• This gives \( f'(1) = -\frac{1}{1^2} \), simplifying to \( -1 \).

Function evaluation is a crucial step in finding the exact slope at the given point, which can have further implications depending on the context of the problem.

Calculus Problems

Calculus problems often start with the task of finding derivatives, but they encompass much more, including analyzing that derivative to gain insights into the original function. Let's turn our attention back to our example, \( f(x) = \frac{1}{x} \).

Here's how you typically tackle calculus problems:

• Identify and rewrite the function, if necessary, to make differentiation straightforward.
• Apply the appropriate differentiation rule, such as the power rule in this case.
• Evaluate the derivative at a specific point to get insights into the function's behavior there.

This methodical approach helps break down complex problems into manageable steps.

In real-world scenarios, such techniques not only reveal the rate of change but can also aid in plotting graphs, optimizing systems, or even solving physics equations. Being comfortable with such calculus problems opens up numerous technical and scientific opportunities.