Math Exercises & Math Problems: Asymptotes of a Function - Free Printable

Actually, there is no specific task or question stated in the image you uploaded — it only lists 26 different functions labeled from a) to Z).

Since no instruction (like “find derivatives”, “simplify”, “find domain”, “graph”, “identify asymptotes”, etc.) is given, I cannot solve a “problem” without knowing what is being asked.

---

✔ However, since this is a common type of problem in calculus or precalculus courses, I’ll make an educated guess:

> Most likely, the task is to find the derivative $ y' $ for each function.

This is a standard exercise when students are learning differentiation rules (quotient rule, product rule, chain rule, logarithmic differentiation, etc.).

---

## 📌 Let’s solve for one example to show you the method, then I can provide all 26 if needed.

---

🔹 Example: Solve for a) $ y = \frac{x}{x + 4} $

This is a quotient → use Quotient Rule:

If $ y = \frac{u}{v} $, then $ y' = \frac{u'v - uv'}{v^2} $

Let:
- $ u = x $ → $ u' = 1 $
- $ v = x + 4 $ → $ v' = 1 $

So,

$$
y' = \frac{(1)(x+4) - (x)(1)}{(x+4)^2} = \frac{x + 4 - x}{(x+4)^2} = \frac{4}{(x+4)^2}
$$

✔ Answer for a): $ \boxed{y' = \frac{4}{(x+4)^2}} $

---

🔹 Example: Solve for b) $ y = \frac{1 - x^2}{x - 2} $

Again, quotient rule.

Let:
- $ u = 1 - x^2 $ → $ u' = -2x $
- $ v = x - 2 $ → $ v' = 1 $

Then:

$$
y' = \frac{(-2x)(x - 2) - (1 - x^2)(1)}{(x - 2)^2}
= \frac{-2x(x - 2) - (1 - x^2)}{(x - 2)^2}
$$

Expand numerator:

$$
-2x^2 + 4x - 1 + x^2 = (-2x^2 + x^2) + 4x - 1 = -x^2 + 4x - 1
$$

✔ Answer for b): $ \boxed{y' = \frac{-x^2 + 4x - 1}{(x - 2)^2}} $

---

🔹 Example: Solve for c) $ y = \frac{2x^2}{2x - 1} $

Quotient rule again.

- $ u = 2x^2 $ → $ u' = 4x $
- $ v = 2x - 1 $ → $ v' = 2 $

$$
y' = \frac{(4x)(2x - 1) - (2x^2)(2)}{(2x - 1)^2}
= \frac{8x^2 - 4x - 4x^2}{(2x - 1)^2}
= \frac{4x^2 - 4x}{(2x - 1)^2}
= \frac{4x(x - 1)}{(2x - 1)^2}
$$

✔ Answer for c): $ \boxed{y' = \frac{4x(x - 1)}{(2x - 1)^2}} $

---

## ✔ If you want ALL 26 derivatives…

I can compute and list them all — just let me know!

But since you didn’t specify the task, here are other possible interpretations:

---

## 🔄 Other Possible Tasks (if not derivatives):

1. Simplify each expression?
- Some can be simplified (e.g., f) $ \frac{x^2 - 9}{1 - x} = \frac{(x-3)(x+3)}{-(x-1)} $, etc.)

2. Find domain?
- For rational functions: exclude where denominator = 0.
- For logs: argument > 0.
- For roots: inside ≥ 0, etc.

3. Find vertical/horizontal asymptotes?

4. Graph each function?

5. Integrate?

---

## 📣 Please Clarify!

To give you the correct solution, please tell me:

> What is the actual task?
> e.g., “Find the derivative of each function”, or “Simplify each expression”, or “Find the domain”, etc.

---

## 💡 Pro Tip:

If this is from a textbook or worksheet, check the heading or instructions above the list — that’s where the real task is usually written.

---

## ✔ For now, I’ll assume you want derivatives — and here’s a quick summary for first 5:

| Function | Derivative |
|----------|------------|
| a) $ \frac{x}{x+4} $ | $ \frac{4}{(x+4)^2} $ |
| b) $ \frac{1-x^2}{x-2} $ | $ \frac{-x^2 + 4x - 1}{(x-2)^2} $ |
| c) $ \frac{2x^2}{2x-1} $ | $ \frac{4x(x - 1)}{(2x - 1)^2} $ |
| d) $ \frac{x^2 + 1}{x} $ | $ \frac{x^2 - 1}{x^2} $ *(simplify first: $ x + \frac{1}{x} $ → $ 1 - \frac{1}{x^2} $)* |
| e) $ \frac{2x^2 - 1 + 3x^3}{3 - 2x^2} $ | Use quotient rule — messy but doable |

---

## 📬 Just say:
> “Please compute the derivative for all 26 functions.”

and I’ll deliver the full list with explanations!

Or specify any other task — I’m ready to help 😊