Calculus worksheet designed to help students identify the correct derivative rule for various functions.

Since you mentioned that you uploaded an image with a task, but I cannot view or access images directly, I will assume that the worksheet you're referring to includes 28 derivative problems, and your goal is to match each problem to one of the 26 basic derivative formulas (numbered 1–26), based on the structure of the function.

You do not need to compute the derivative, just identify which basic derivative rule applies to each function.

Let me go through each problem and determine which basic derivative formula it corresponds to. To do this, I’ll use common standard derivative rules such as:

- Power Rule
- Product Rule
- Quotient Rule
- Chain Rule
- Derivatives of trig functions
- Exponential and logarithmic derivatives
- Inverse trig functions
- Logarithmic differentiation

We’ll assign the number from the Derivative Formulas sheet (which I assume contains standard rules like):

1. $ \frac{d}{dx}[x^n] = nx^{n-1} $
2. $ \frac{d}{dx}[c] = 0 $
3. $ \frac{d}{dx}[cf(x)] = c f'(x) $
4. $ \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) $
5. $ \frac{d}{dx}[f(x)g(x)] = f'g + fg' $ (Product Rule)
6. $ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'g - fg'}{g^2} $ (Quotient Rule)
7. $ \frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) $ (Chain Rule)
8. $ \frac{d}{dx}[e^x] = e^x $
9. $ \frac{d}{dx}[e^{u(x)}] = e^{u(x)} u'(x) $
10. $ \frac{d}{dx}[\ln x] = \frac{1}{x} $
11. $ \frac{d}{dx}[\ln u(x)] = \frac{u'(x)}{u(x)} $
12. $ \frac{d}{dx}[\sin x] = \cos x $
13. $ \frac{d}{dx}[\cos x] = -\sin x $
14. $ \frac{d}{dx}[\tan x] = \sec^2 x $
15. $ \frac{d}{dx}[\cot x] = -\csc^2 x $
16. $ \frac{d}{dx}[\sec x] = \sec x \tan x $
17. $ \frac{d}{dx}[\csc x] = -\csc x \cot x $
18. $ \frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1 - x^2}} $
19. $ \frac{d}{dx}[\cos^{-1} x] = -\frac{1}{\sqrt{1 - x^2}} $
20. $ \frac{d}{dx}[\tan^{-1} x] = \frac{1}{1 + x^2} $
21. $ \frac{d}{dx}[\cot^{-1} x] = -\frac{1}{1 + x^2} $
22. $ \frac{d}{dx}[a^x] = a^x \ln a $
23. $ \frac{d}{dx}[x^x] = x^x (\ln x + 1) $ (logarithmic differentiation)
24. $ \frac{d}{dx}[a^u(x)] = a^{u(x)} \ln a \cdot u'(x) $
25. $ \frac{d}{dx}[u(x)^{v(x)}] $ → log diff (use implicit)
26. $ \frac{d}{dx}[\sin(u(x))] = \cos(u(x)) \cdot u'(x) $ — general chain rule for trig

Note: The actual numbering may vary slightly depending on the sheet, but these are typical.

Now let’s match each problem to the appropriate rule.

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1. $ y = 8x $

- This is a linear function: constant times $ x $
- Use Power Rule or Rule 1: $ \frac{d}{dx}[x^n] = nx^{n-1} $, here $ n=1 $
- Also involves constant multiple → Rule 3 might be involved
- But simplest: Rule 1 (Power Rule)

✔ Answer: 1

---

2. $ y = \sin^3(2x) $

- This means $ (\sin(2x))^3 $
- Outer function: $ u^3 $, inner: $ \sin(2x) $
- So: chain rule applied twice
- First: power rule on $ u^3 $, then derivative of $ \sin(2x) $, then derivative of $ 2x $
- So chain rule + trig derivative
- Likely matches Rule 26 (derivative of $ \sin(u) $), but also involves power rule and chain rule

But since it's $ [\sin(2x)]^3 $, we can think of it as $ u^3 $ where $ u = \sin(2x) $, so:
- Power rule on outer: $ 3u^2 u' $
- Then $ u' = \cos(2x) \cdot 2 $
- So overall: Power Rule + Chain Rule + Trig Rule

But if the list has a "Power of a function" rule, it might be Rule 7 (Chain Rule), but more specifically, this would involve Rule 26 for $ \sin $, and Rule 1 for power.

But since it's a composition of powers and trig, likely Rule 7 (Chain Rule) is the main one.

But perhaps there's a specific rule for $ (\text{trig})^n $. However, in most sheets, this is handled by Chain Rule.

Alternatively, if the sheet has a rule like:

> $ \frac{d}{dx}[f(u)^n] = n f(u)^{n-1} \cdot f'(u) \cdot u' $

That’s Chain Rule, so Rule 7

✔ Answer: 7

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3. $ y = x e^{\cos x} $

- Product of two functions: $ x $ and $ e^{\cos x} $
- So Product Rule (Rule 5)
- Also, $ e^{\cos x} $ requires Chain Rule (Rule 9)

So primary rule: Product Rule

✔ Answer: 5

---

4. $ y = \sin^{-1}(x^3) $

- Inverse sine of a function
- Rule: $ \frac{d}{dx}[\sin^{-1}(u)] = \frac{1}{\sqrt{1 - u^2}} \cdot u' $
- So Rule 18 (if Rule 18 is inverse sine)

✔ Answer: 18

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5. $ y = \sin(\ln(x^3)) $

- $ \sin(u) $, where $ u = \ln(x^3) $
- So chain rule: derivative of $ \sin(u) $ times derivative of $ \ln(x^3) $
- $ \ln(x^3) = 3\ln x $, so derivative is $ \frac{3}{x} $
- So overall: Chain Rule with trig and log
- Mainly Rule 26 (derivative of $ \sin(u) $)

✔ Answer: 26

---

6. $ y = \ln(x^2 - 5) $

- Natural log of a function: $ \ln(u) $, $ u = x^2 - 5 $
- So derivative: $ \frac{1}{u} \cdot u' $
- This is Rule 11 (derivative of $ \ln u $)

✔ Answer: 11

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7. $ y = 5 $

- Constant function
- Derivative is zero → Rule 2

✔ Answer: 2

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8. $ y = \tan(3x^2) $

- Tangent of a function: $ \tan(u) $, $ u = 3x^2 $
- Derivative: $ \sec^2(u) \cdot u' $
- So Rule 14 (derivative of $ \tan u $) or Chain Rule

If Rule 14 is $ \frac{d}{dx}[\tan u] = \sec^2 u \cdot u' $, then it's Rule 14

✔ Answer: 14

---

9. $ y = 3x^{-1} + 12x $

- Sum of two terms: $ 3/x $ and $ 12x $
- Each term uses Power Rule (Rule 1)
- Also constant multiple (Rule 3), sum rule (Rule 4)
- But main rule: Power Rule

✔ Answer: 1

---

10. $ y = e^{\arcsin x} $

- Exponential with exponent being $ \arcsin x $
- So $ e^{u(x)} $, $ u = \arcsin x $
- Derivative: $ e^u \cdot u' $
- So Rule 9: $ \frac{d}{dx}[e^{u}] = e^u u' $

✔ Answer: 9

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11. $ y = \tan^{-1}(\ln x) $

- Inverse tangent of a function
- $ \frac{d}{dx}[\tan^{-1}(u)] = \frac{1}{1+u^2} \cdot u' $
- So Rule 20 (derivative of $ \tan^{-1} u $)

✔ Answer: 20

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12. $ y = \sec^2(3x) $

- This is $ [\sec(3x)]^2 $
- So outer: square, inner: $ \sec(3x) $
- Use Chain Rule and derivative of secant
- $ \frac{d}{dx}[\sec^2 u] = 2\sec u \cdot (\sec u \tan u) \cdot u' $
- So involves Rule 16 (derivative of $ \sec x $) and Chain Rule

But since it's a composite function, likely Rule 7 (Chain Rule) is the base

However, if there’s a rule for $ \sec^2 x $, it might be listed separately.

But typically, this is handled via Chain Rule.

Alternatively, if the sheet has a rule for $ \frac{d}{dx}[\sec^2 u] $, but more likely, it's Rule 7

Wait — actually, $ \sec^2(3x) $ is the derivative of $ \tan(3x) $, but we’re differentiating it.

But anyway, derivative is $ 2\sec(3x)\cdot \sec(3x)\tan(3x)\cdot 3 = 6\sec^2(3x)\tan(3x) $

So it uses Chain Rule and derivative of secant

But the main rule is Chain Rule applied to $ [f(x)]^2 $

So Rule 7 (Chain Rule)

✔ Answer: 7

---

13. $ y = \cot x $

- Basic derivative of cotangent
- $ \frac{d}{dx}[\cot x] = -\csc^2 x $
- So Rule 15

✔ Answer: 15

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14. $ y = x \cos\left(\frac{1}{x}\right) $

- Product of $ x $ and $ \cos(1/x) $
- So Product Rule (Rule 5)
- And $ \cos(1/x) $ needs Chain Rule (since $ u = 1/x $)

So main rule: Product Rule

✔ Answer: 5

---

15. $ y = (e^2)^x $

- Note: $ e^2 $ is a constant, so $ (e^2)^x = e^{2x} $
- So $ y = e^{2x} $, derivative: $ 2e^{2x} $
- This is Rule 9: derivative of $ e^{u(x)} $, $ u = 2x $

✔ Answer: 9

---

16. $ y = e^{3x} $

- Again, exponential function: $ e^{u(x)} $, $ u = 3x $
- So Rule 9

✔ Answer: 9

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17. $ y = e^e $

- $ e $ is a constant, so $ e^e $ is a constant
- Derivative is 0 → Rule 2

✔ Answer: 2

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18. $ y = \frac{x^2}{e^{2x}} $

- Quotient: numerator $ x^2 $, denominator $ e^{2x} $
- So Quotient Rule (Rule 6)

✔ Answer: 6

---

19. $ y = \ln(x)\sin(x) $

- Product of $ \ln x $ and $ \sin x $
- So Product Rule (Rule 5)

✔ Answer: 5

---

20. $ y = \ln(\sin x) $

- $ \ln(u) $, $ u = \sin x $
- So derivative: $ \frac{1}{\sin x} \cdot \cos x = \cot x $
- So Rule 11: derivative of $ \ln u $

✔ Answer: 11

---

21. $ y = \tan^2(\ln(3x^2 + 3) + 8x) $

- This is $ [\tan(u)]^2 $, where $ u = \ln(3x^2 + 3) + 8x $
- So outer: square, middle: tan, inner: $ \ln(3x^2 + 3) + 8x $
- So multiple layers: Chain Rule applied repeatedly
- But the outermost is power of a function → Rule 7 (Chain Rule)

✔ Answer: 7

---

22. $ y = x^\pi $

- Power function: $ x^n $, $ n = \pi $ (constant)
- So Power Rule (Rule 1)

✔ Answer: 1

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23. $ y = \pi^x $

- $ \pi^x $: exponential with base $ \pi $
- Derivative: $ \pi^x \ln \pi $
- So Rule 22: $ \frac{d}{dx}[a^x] = a^x \ln a $

✔ Answer: 22

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24. $ y = \pi^\pi $

- $ \pi $ is constant, so $ \pi^\pi $ is constant
- Derivative = 0 → Rule 2

✔ Answer: 2

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25. $ y = x^x $

- Variable base and variable exponent
- Requires logarithmic differentiation
- This is Rule 23 (if present): $ \frac{d}{dx}[x^x] = x^x (\ln x + 1) $

✔ Answer: 23

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26. $ y = \frac{\sin(3x)}{\sqrt{x^2 - 5}} $

- Quotient: numerator $ \sin(3x) $, denominator $ (x^2 - 5)^{1/2} $
- So Quotient Rule (Rule 6)

✔ Answer: 6

---

27. $ y = \cot^{-1}\left(\frac{1}{1+x^2}\right) $

- Inverse cotangent of a function
- $ \frac{d}{dx}[\cot^{-1}(u)] = -\frac{1}{1+u^2} \cdot u' $
- So Rule 21 (derivative of $ \cot^{-1} u $)

✔ Answer: 21

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28. $ y = 3x^2 - \cos^{-1}x $

- Difference of two functions: $ 3x^2 $ and $ \cos^{-1}x $
- So Sum/Difference Rule (Rule 4)
- Also, $ \cos^{-1}x $ derivative is $ -\frac{1}{\sqrt{1 - x^2}} $ → Rule 19

But since it's a difference, main rule is Rule 4 (sum/difference)

But if the sheet lists derivative of $ \cos^{-1}x $ as a separate rule, it might be Rule 19

But the question asks for the basic derivative formula used.

The function has two parts: polynomial and inverse cosine.

So the overall structure is sum/difference, so Rule 4

But if they want the rule for $ \cos^{-1}x $, it's Rule 19

But since the whole expression is a difference, and the derivative involves both, but the main rule applied is Rule 4

However, often in such worksheets, if a function has multiple components, they expect the primary rule for the non-trivial part.

But here, $ \cos^{-1}x $ is the non-trivial one.

But since it's subtraction, and $ 3x^2 $ uses Power Rule, $ \cos^{-1}x $ uses Rule 19.

But the formula for $ \cos^{-1}x $ is Rule 19

So likely, the intended answer is Rule 19, because $ \cos^{-1}x $ is not trivial.

But strictly speaking, the operation is subtraction, so Rule 4

But given the context, probably they want the rule for the inverse cosine part.

Let’s see: if $ y = f(x) - g(x) $, then $ y' = f'(x) - g'(x) $, so Rule 4 applies.

But if the problem is about identifying the type of derivative, then the non-polynomial part uses Rule 19

But since the instruction says "place the number of the basic derivative formula", and Rule 19 is a basic formula, while Rule 4 is also basic.

But in many such worksheets, Rule 4 is for sum/difference, and it's used when combining functions.

But here, the key is that $ \cos^{-1}x $ requires its own rule.

So likely, the answer is Rule 19, because that's the non-obvious part.

But let's check: the function is $ 3x^2 - \cos^{-1}x $, so derivative is $ 6x + \frac{1}{\sqrt{1 - x^2}} $

So two parts: Power Rule and Inverse Cosine Rule

But the formula for $ \cos^{-1}x $ is Rule 19

✔ Answer: 19

---

Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | 1 |
| 2 | 7 |
| 3 | 5 |
| 4 | 18 |
| 5 | 26 |
| 6 | 11 |
| 7 | 2 |
| 8 | 14 |
| 9 | 1 |
| 10 | 9 |
| 11 | 20 |
| 12 | 7 |
| 13 | 15 |
| 14 | 5 |
| 15 | 9 |
| 16 | 9 |
| 17 | 2 |
| 18 | 6 |
| 19 | 5 |
| 20 | 11 |
| 21 | 7 |
| 22 | 1 |
| 23 | 22 |
| 24 | 2 |
| 25 | 23 |
| 26 | 6 |
| 27 | 21 |
| 28 | 19 |

---

Summary of Matches:

- Rule 1 (Power Rule): 1, 9, 22
- Rule 2 (Constant): 7, 17, 24
- Rule 5 (Product Rule): 3, 14, 19
- Rule 6 (Quotient Rule): 18, 26
- Rule 7 (Chain Rule): 2, 12, 21
- Rule 9 (Derivative of $ e^u $): 10, 15, 16
- Rule 11 (Derivative of $ \ln u $): 6, 20
- Rule 14 (Derivative of $ \tan u $): 8
- Rule 15 (Derivative of $ \cot x $): 13
- Rule 18 (Derivative of $ \sin^{-1} u $): 4
- Rule 19 (Derivative of $ \cos^{-1} x $): 28
- Rule 20 (Derivative of $ \tan^{-1} u $): 11
- Rule 21 (Derivative of $ \cot^{-1} u $): 27
- Rule 22 (Derivative of $ a^x $): 23
- Rule 23 (Derivative of $ x^x $): 25

Note: Some rules like Rule 26 (derivative of $ \sin u $) were used in #5, but not explicitly listed above. But in our assignment, we used Rule 26 for #5.

Wait — earlier I said #5: $ y = \sin(\ln(x^3)) $ → Rule 26

But in my final table, I didn’t include it.

Let me fix that.

In Problem 5, we used Rule 26 (derivative of $ \sin(u) $)

Similarly, Problem 2 used Rule 26? No, wait:

Actually, in Problem 2: $ y = \sin^3(2x) = [\sin(2x)]^3 $

This is not $ \sin(u) $, but $ u^3 $, so Chain Rule (Rule 7), not Rule 26.

So Rule 26 was used only in #5

But in my initial analysis, I said #5 → Rule 26

But in the final table, I have no entry for Rule 26.

Let me correct:

- Problem 5: $ y = \sin(\ln(x^3)) $ → derivative of $ \sin(u) $ → Rule 26

So #5 → 26

But in my final table, I had #5 → 26? Wait, yes — in the list above, I wrote:

> 5. $ y = \sin(\ln(x^3)) $ → ✔ Answer: 26

Yes, it's there.

So all good.

---

Final Answer List:

```
1. 1
2. 7
3. 5
4. 18
5. 26
6. 11
7. 2
8. 14
9. 1
10. 9
11. 20
12. 7
13. 15
14. 5
15. 9
16. 9
17. 2
18. 6
19. 5
20. 11
21. 7
22. 1
23. 22
24. 2
25. 23
26. 6
27. 21
28. 19
```

These are the numbers of the basic derivative formulas corresponding to each problem.

Let me know if you'd like the actual derivative formulas or explanations for any!