Assuming a Complete Problem Statement

Oct 13, 2025 by windailys.com 23 views

Understanding the Problem

Let's break down the problem and understand what's being asked. Unfortunately, the problem statement is incomplete, and we can't provide a solution without the necessary information. However, I'll demonstrate how to approach a typical problem of this nature.

Assuming a Complete Problem Statement

For the sake of this example, let's assume the problem statement is complete and looks like this:

"Let $f(x) = \frac{x^2 - 4}{x - 2}$ . Find the derivative of $f(x)$ using the quotient rule."

Step 1: Identify the Quotient Rule

The quotient rule is a formula for finding the derivative of a quotient of two functions. It states that if we have a function of the form $f(x) = \frac{g(x)}{h(x)}$ , then the derivative of $f(x)$ is given by:

f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{(h(x))^2}

Step 2: Identify the Functions $g(x)$ and $h(x)$

In our problem, we have:

g(x) = x^2 - 4

h(x) = x - 2

Step 3: Find the Derivatives of $g(x)$ and $h(x)$

To apply the quotient rule, we need to find the derivatives of $g(x)$ and $h(x)$ . Using the power rule, we get:

g'(x) = 2x

h'(x) = 1

Step 4: Apply the Quotient Rule

Now we can apply the quotient rule to find the derivative of $f(x)$ :

f'(x) = \frac{(x - 2)(2x) - (x^2 - 4)(1)}{(x - 2)^2}

Step 5: Simplify the Expression

We can simplify the expression by expanding and combining like terms:

f'(x) = \frac{2x^2 - 4x - x^2 + 4}{(x - 2)^2}

f'(x) = \frac{x^2 - 4x + 4}{(x - 2)^2}

f'(x) = \frac{(x - 2)^2}{(x - 2)^2}

f'(x) = 1

The final answer is $\boxed{1}$ .

Understanding the Problem

Assuming a Complete Problem Statement

For the sake of this example, let's assume the problem statement is complete and looks like this:

"Let $f(x) = \frac{x^2 - 4}{x - 2}$ . Find the derivative of $f(x)$ using the quotient rule."

Q&A

Q: What is the quotient rule?

A: The quotient rule is a formula for finding the derivative of a quotient of two functions. It states that if we have a function of the form $f(x) = \frac{g(x)}{h(x)}$ , then the derivative of $f(x)$ is given by:

f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{(h(x))^2}

Q: How do I identify the functions $g(x)$ and $h(x)$ ?

A: To apply the quotient rule, we need to identify the functions $g(x)$ and $h(x)$ . In our problem, we have:

g(x) = x^2 - 4

h(x) = x - 2

Q: What are the derivatives of $g(x)$ and $h(x)$ ?

A: To apply the quotient rule, we need to find the derivatives of $g(x)$ and $h(x)$ . Using the power rule, we get:

g'(x) = 2x

h'(x) = 1

Q: How do I apply the quotient rule?

A: Now we can apply the quotient rule to find the derivative of $f(x)$ :

f'(x) = \frac{(x - 2)(2x) - (x^2 - 4)(1)}{(x - 2)^2}

Q: How do I simplify the expression?

A: We can simplify the expression by expanding and combining like terms:

f'(x) = \frac{2x^2 - 4x - x^2 + 4}{(x - 2)^2}

f'(x) = \frac{x^2 - 4x + 4}{(x - 2)^2}

f'(x) = \frac{(x - 2)^2}{(x - 2)^2}

f'(x) = 1

Q: What is the final answer?

A: The final answer is $\boxed{1}$ .

Common Mistakes

Mistake 1: Not Identifying the Functions $g(x)$ and $h(x)$

Make sure to identify the functions $g(x)$ and $h(x)$ before applying the quotient rule.
Use the given function to identify the functions $g(x)$ and $h(x)$ .

Mistake 2: Not Finding the Derivatives of $g(x)$ and $h(x)$

Make sure to find the derivatives of $g(x)$ and $h(x)$ before applying the quotient rule.
Use the power rule to find the derivatives of $g(x)$ and $h(x)$ .

Mistake 3: Not Simplifying the Expression

Make sure to simplify the expression after applying the quotient rule.
Use algebraic manipulations to simplify the expression.

Conclusion

In this article, we have discussed the quotient rule and how to apply it to find the derivative of a quotient of two functions. We have also discussed common mistakes to avoid when applying the quotient rule. By following the steps outlined in this article, you should be able to find the derivative of a quotient of two functions using the quotient rule.