https://file.aiquickdraw.com/mathbot/file/1726458764994u02ilrga.jpg

The problem asks to find the slope of the graph of the given function at a specific point. The function provided is:

\[
y = \frac{4}{(x + 2)^2}
\]

The task is to calculate the derivative \( y'(x) \) and then evaluate it at \( x = 0 \) to determine the slope at the point \((0, 1)\).

### Step-by-step solution:

1. **Rewrite the function** in a form that's easier to differentiate:
   \[
   y = 4(x + 2)^{-2}
   \]

2. **Differentiate** the function using the chain rule. The general rule for the derivative of \( (xThe derivative of the function \( y = \frac{4}{(x + 2)^2} \) is:

\[
y'(x) = \frac{-8}{(x + 2)^3}
\]

Now, evaluating this derivative at \( x = 0 \) gives:

\[
y'(0) = \frac{-8}{(0 + 2)^3} = -1
\]

So, the slope of the graph at the point \((0, 1)\) is \( \boxed{-1} \).

Would you like further details on any part of the solution?

Here are five related questions to explore:

1. How is the chain rule applied when differentiating functions with powers?
2. How do we interpret the slope of a curve at a point geometrically?
3. Can the derivative ever be undefined for this function?
4. What is the significance of negative slopes in the context of graphing?
5. How would you approach differentiating more complex rational functions?

**Tip**: When differentiating, remember to rewrite powers and roots in their exponential form for easier application of rules like the chain rule.

Find the slope of the graph of the function y = 4 / (x + 2)^2 at the point (0,1).

Learn how to find the derivative of the function y = 4 / (x + 2)^2 using the chain rule, and evaluate the slope at the point (0,1). Suitable for Calculus 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