https://file.aiquickdraw.com/mathbot/file/17254523207204rz3dd3n.jpg

We are asked to differentiate the function \( y = \frac{2}{x - 3} \) with respect to \( x \).

### Differentiation process:

The given function is a rational function. We can rewrite it as:
\[
y = 2(x - 3)^{-1}
\]

Now, we will differentiate it using the power rule and chain rule.

Let’s compute:
\[
\frac{dy}{dx} = \frac{d}{dx} \left[ 2(x - 3)^{-1} \right]
\]
Using the chain rule:
\[
\frac{dy}{dx} = 2 \cdot (-1)(x - 3)^{-2} \cdot \frac{d}{dx}(x - 3)
\]
Since \( \frac{d}{dx}(x - 3) = 1 \), we get:
\[
\frac{dy}{dx} = -2(x - 3)^{-2}
\]

Thus, the derivative is:
\[
\boxed{\frac{dy}{dx} = \frac{-2}{(x - 3)^2}}
\]

Let me know if you would like further clarification or have any questions!

### Follow-up questions:
1. What is the chain rule, and why is it applied here?
2. How can you verify if your derivative is correct?
3. How would the derivative change if the numerator were a function of \( x \)?
4. What is the physical interpretation of the derivative in this context?
5. Can you sketch the graph of the function and its derivative?

#### Tip:
When dealing with rational functions, try rewriting them in a form suitable for applying the chain rule or the quotient rule directly.

Differentiate the following with respect to x: y = 2 / (x - 3)

Learn how to differentiate the rational function y = 2 / (x - 3) with respect to x. This problem involves the application of the chain rule and power rule, key concepts in calculus. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation