Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of each function. \(f(x)=\frac{1}{x-2}\)

Step-by-Step Solution

Answer

The short answer for the derivative of \(f(x)=\frac{1}{x-2}\) is \(f'(x) = -\frac{1}{(x-2)^2}\).

Step 1: Rewrite the function with a negative exponent

We first rewrite the given function as an expression with a negative exponent by moving the denominator to the numerator: \[f(x)=(x-2)^{-1}\]

Step 2: Apply the power rule for differentiation

Now we can apply the power rule for differentiation, which states that the derivative of \(x^n\) is \(nx^{n-1}\). In this case, \(n=-1\). So we have: \[f'(x) = -1(x-2)^{-1-1}\]

Step 3: Simplify the derivative

Finally, we simplify the derivative by rewriting it as a fraction: \[f'(x) = -1(x-2)^{-2}\] \[f'(x) = -\frac{1}{(x-2)^2}\] So the derivative of the given function is: \[f'(x) = -\frac{1}{(x-2)^2}\]

Key Concepts

Understanding the Power Rule

The power rule is a fundamental concept in calculus which simplifies the process of differentiation. Differentiation is essentially finding the rate at which a function changes. The power rule is specifically used when you have functions that are powers of a variable, such as \(x^n\). Here’s the essence of the power rule:

• When you have a function in the form of \(x^n\), the derivative (which tells you the rate of change) is found by multiplying the power \(n\) by \(x\), and then subtracting one from the power. So, for \(x^n\), the derivative is \(nx^{n-1}\).

This rule significantly streamlines the differentiation process because it provides a straightforward formula, making it easy to apply to functions that involve terms like \(x^3, x^{-2},\) etc. In the exercise, we used the power rule by recognizing the function \((x-2)^{-1}\), and then applying \(n = -1\) to find the derivative. Basically, with the power rule, dealing with differentiations becomes less complex and manageable.

Working with Negative Exponents

Negative exponents can initially seem daunting, but they simply indicate the reciprocal of a base raised to a positive exponent. For instance, \(x^{-n}\) is another way of writing \(\frac{1}{x^n}\). In the context of calculus and differentiation, negative exponents often appear when you have rational functions or need to rewrite a denominator in the numerator.

• A negative exponent allows us to utilize rules like the power rule more straightforwardly without having fractions muddle the differentiation process.
• For example, \((x-2)^{-1}\) means \(\frac{1}{x-2}\). By converting it to this form with a negative exponent, it becomes easier to apply the power rule.

Recognizing and manipulating negative exponents is key when simplifying expressions and solving derivative problems, as seen in the step-by-step solution provided. Mastering them ensures you can rearrange and differentiate expressions effectively.

Exploring Rational Functions

Rational functions are an excellent place to apply those calculus skills. These functions are defined as the ratio of two polynomials, such as \(\frac{1}{x-2}\), which you’ve worked on in the exercise. Here's what you need to know about them:

• Rational functions can often be rewritten using negative exponents, making them easier to differentiate using the power rule.
• While differentiating rational functions, you might encounter fractions that need simplification, such as flipping denominators into numerators using negative exponents.
• They often require additional simplification after finding a derivative, as seen where we simplified \(-1(x-2)^{-2}\) to \(-\frac{1}{(x-2)^2}\).

Understanding how to work with rational functions and how they interact with differentiation is crucial. Often, when you see a fraction-like \(\frac{1}{x-a}\), you can rewrite it to take advantage of known rules like the power rule, ultimately simplifying your calculus work by streamlining derivative calculations.