https://file.aiquickdraw.com/mathbot/file/1730029560309za4a2z67.jpg

Let's solve question 16 from the image.

We are asked to **differentiate the function** \( y = \frac{2x + 5}{2 - x} \) and **find the slope of the tangent line** at \( x = -2 \).

---

### Step 1: Differentiate the function
The function is a quotient, so we will apply the **quotient rule**. The quotient rule states:
\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]
Here:
- \( u(x) = 2x + 5 \)
- \( v(x) = 2 - x \)

First, calculate the derivatives of \( u(x) \) and \( v(x) \):
\[
u'(x) = \frac{d}{dx}(2x + 5) = 2
\]
\[
v'(x) = \frac{d}{dx}(2 - x) = -1
\]

Now, apply the quotient rule:
\[
y'(x) = \frac{2(2 - x) - (2x + 5)(-1)}{(2 - x)^2}
\]
Simplify the numerator:
\[
y'(x) = \frac{4 - 2x + 2x + 5}{(2 - x)^2}
\]
\[
y'(x) = \frac{9}{(2 - x)^2}
\]

---

### Step 2: Evaluate the derivative at \( x = -2 \)
We substitute \( x = -2 \) into the derivative \( y'(x) = \frac{9}{(2 - x)^2} \).

\[
y'(-2) = \frac{9}{(2 - (-2))^2} = \frac{9}{(2 + 2)^2} = \frac{9}{4^2} = \frac{9}{16}
\]

---

### Final Answer
The slope of the tangent line at \( x = -2 \) is:
\[
\frac{9}{16}
\]

---

Do you want further clarification, or would you like to see the equation of the tangent line at \( x = -2 \)? 

---

**Related questions:**
1. How is the quotient rule derived?
2. What happens if we reverse the roles of \( u(x) \) and \( v(x) \) in the quotient rule?
3. How do we find the equation of the tangent line given the slope?
4. What are other ways to compute the derivative besides the quotient rule?
5. Can you apply the product rule instead of the quotient rule here?

**Tip:** Always simplify the function before differentiating if possible. This can make the derivative easier to compute.

Learn how to differentiate the function y = (2x + 5)/(2 - x) and find the slope of the tangent line at x = -2. This solution includes a detailed application of the quotient rule and step-by-step calculations. 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