Worksheet for Derivative Formulas - Free Printable

Since I can't view or access uploaded images directly, I’ll help you solve this Derivative Formulas Worksheet based on the text you've provided. You're asked to match each derivative problem with the number of the basic derivative formula (from a reference sheet numbered 1–26), without actually solving the derivatives.

However, since I don’t have the actual "sheet of Derivative Formulas" (which would list formulas like:
- 1. $ \frac{d}{dx}[x^n] = nx^{n-1} $
- 2. $ \frac{d}{dx}[e^x] = e^x $
- 3. $ \frac{d}{dx}[\ln x] = \frac{1}{x} $, etc.), I will assume a standard list of basic derivative rules commonly used in calculus, and assign the most likely formula numbers based on typical numbering systems.

Let’s go through each problem and identify which basic derivative rule applies.

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🔢 Assumed Standard Derivative Formula List (1–26):

Here is a common order for such sheets:

1. $ \frac{d}{dx}[x^n] = nx^{n-1} $
2. $ \frac{d}{dx}[c] = 0 $ (constant)
3. $ \frac{d}{dx}[e^x] = e^x $
4. $ \frac{d}{dx}[a^x] = a^x \ln a $
5. $ \frac{d}{dx}[\ln x] = \frac{1}{x} $
6. $ \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a} $
7. $ \frac{d}{dx}[\sin x] = \cos x $
8. $ \frac{d}{dx}[\cos x] = -\sin x $
9. $ \frac{d}{dx}[\tan x] = \sec^2 x $
10. $ \frac{d}{dx}[\cot x] = -\csc^2 x $
11. $ \frac{d}{dx}[\sec x] = \sec x \tan x $
12. $ \frac{d}{dx}[\csc x] = -\csc x \cot x $
13. $ \frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1-x^2}} $
14. $ \frac{d}{dx}[\cos^{-1} x] = -\frac{1}{\sqrt{1-x^2}} $
15. $ \frac{d}{dx}[\tan^{-1} x] = \frac{1}{1+x^2} $
16. $ \frac{d}{dx}[\cot^{-1} x] = -\frac{1}{1+x^2} $
17. $ \frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) $ (Chain Rule)
18. $ \frac{d}{dx}[f(x)g(x)] = f'g + fg' $ (Product Rule)
19. $ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'g - fg'}{g^2} $ (Quotient Rule)
20. $ \frac{d}{dx}[u^n] = n u^{n-1} u' $ (Power Rule with Chain)
21. $ \frac{d}{dx}[e^{u}] = e^{u} u' $
22. $ \frac{d}{dx}[\ln u] = \frac{1}{u} u' $
23. $ \frac{d}{dx}[a^u] = a^u \ln a \cdot u' $
24. $ \frac{d}{dx}[u^v] = v u^{v-1} u' + u^v \ln u \cdot v' $ (Logarithmic Differentiation)
25. $ \frac{d}{dx}[x^x] = x^x(1 + \ln x) $ (Special case of 24)
26. $ \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) $ (Sum/Difference Rule)

> Note: Some of these are general rules, others are specific functions. The exact numbering may vary slightly, but we'll match as closely as possible.

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Now let's analyze each problem and assign the most likely formula number from above.

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✔ Problem Solutions (Matching Formula Numbers)

1. $ y = 8x $
→ Power rule: $ \frac{d}{dx}[x^n] $, $ n=1 $
→ Formula #1

2. $ y = \sin^3(2x) $
→ This is $ [\sin(2x)]^3 $, so chain rule + power rule
→ Formula #20 (Power rule with chain) or #17 (Chain Rule)
→ But since it's a composite function raised to a power, best fit: #20

3. $ y = x e^{\cos x} $
→ Product of two functions: $ x $ and $ e^{\cos x} $
→ Formula #18 (Product Rule)

4. $ y = \sin^{-1}(x^3) $
→ Inverse sine of a function → chain rule applied to arcsin
→ Formula #13 (derivative of $ \sin^{-1} u $)
→ So: #13

5. $ y = \sin(\ln(x^3)) $
→ Composition: sin of ln of x³ → chain rule
→ Formula #17 (Chain Rule)

6. $ y = \ln(x^2 - 5) $
→ Logarithm of a function → Formula #22 (Derivative of $ \ln u $)

7. $ y = 5 $
→ Constant → Formula #2

8. $ y = \tan(3x^2) $
→ Tangent of a function → chain rule
→ Formula #17 (Chain Rule) — or specifically, derivative of tan(u) → use #9 with chain

But since the base rule is $ \frac{d}{dx}[\tan u] = \sec^2 u \cdot u' $, and that’s not listed separately, we use #17 (Chain Rule)

→ #17

9. $ y = 3x^{-1} + 12x $
→ Sum of powers → use #1 (power rule) and #26 (sum rule)
→ But since both terms are polynomials, main rule is #1, and sum rule applies
→ Best: #1 (and #26 implicitly), but since only one number per blank, choose #1

However, if #26 is sum/difference rule, then this uses #26 and #1. But since only one number is allowed, and the structure is polynomial, likely #1 is expected.

→ #1

10. $ y = e^{x \sin x} $
→ Exponential of a product → use #21: $ \frac{d}{dx}[e^u] = e^u u' $
→ So: #21

11. $ y = \tan^{-1}(\ln x) $
→ Arctangent of a function → chain rule
→ Formula #15 (derivative of $ \tan^{-1} u $)
→ So: #15

12. $ y = \sec^2(3x) $
→ This is $ [\sec(3x)]^2 $ → chain rule + power rule
→ Formula #20 (Power rule with chain)

13. $ y = \cot x $
→ Derivative of cotangent → Formula #10

14. $ y = x \cos\left(\frac{1}{x}\right) $
→ Product of $ x $ and $ \cos(1/x) $ → #18 (Product Rule)

15. $ y = (e^2)^x $
→ $ (e^2)^x = e^{2x} $, so exponential with base $ e $, exponent $ 2x $
→ Use $ \frac{d}{dx}[e^{u}] = e^u u' $ → #21

16. $ y = e^{3x} $
→ Same as above → #21

17. $ y = e^e $
→ $ e^e $ is a constant (since $ e $ is constant)
→ Derivative is zero → #2

18. $ y = \frac{x^2}{e^{2x}} $
→ Quotient of two functions → #19 (Quotient Rule)

19. $ y = \ln(x)\sin(x) $
→ Product of two functions → #18 (Product Rule)

20. $ y = \ln(\sin x) $
→ Log of a function → #22 (Derivative of $ \ln u $)

21. $ y = \tan^5(\ln(3x^2 + 3) + 8x) $
→ $ [\tan(\text{something})]^5 $ → Power rule + chain rule
→ #20 (Power rule with chain)

22. $ y = x^\pi $
→ Power function with constant exponent → #1 (Power rule)

23. $ y = \pi^x $
→ Exponential with constant base $ \pi $ → #4 ($ a^x $ derivative)

24. $ y = \pi^\pi $
→ Constant → #2

25. $ y = x^x $
→ Variable base and exponent → #24 (Logarithmic differentiation)

26. $ y = \frac{\sin(3x)}{\sqrt{x^2 - 5}} $
→ Quotient of two functions → #19 (Quotient Rule)

27. $ y = \cot^{-1}\left(\frac{1}{1+x^2}\right) $
→ Inverse cotangent of a function → #16 (derivative of $ \cot^{-1} u $)

28. $ y = 3x^2 - \cos^{-1}x $
→ Difference of two functions → #26 (Sum/Difference Rule)
→ Also involves derivative of $ \cos^{-1}x $ → #14, but overall structure is sum/difference
→ Since only one number per blank, and the form is $ f(x) \pm g(x) $, best choice: #26

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✔ Final Answer Key (Fill-in-the-blanks):

| Problem | Formula # |
|--------|-----------|
| 1. $ y = 8x $ | 1 |
| 2. $ y = \sin^3(2x) $ | 20 |
| 3. $ y = x e^{\cos x} $ | 18 |
| 4. $ y = \sin^{-1}(x^3) $ | 13 |
| 5. $ y = \sin(\ln(x^3)) $ | 17 |
| 6. $ y = \ln(x^2 - 5) $ | 22 |
| 7. $ y = 5 $ | 2 |
| 8. $ y = \tan(3x^2) $ | 17 |
| 9. $ y = 3x^{-1} + 12x $ | 1 |
|10. $ y = e^{x \sin x} $ | 21 |
|11. $ y = \tan^{-1}(\ln x) $ | 15 |
|12. $ y = \sec^2(3x) $ | 20 |
|13. $ y = \cot x $ | 10 |
|14. $ y = x \cos(1/x) $ | 18 |
|15. $ y = (e^2)^x $ | 21 |
|16. $ y = e^{3x} $ | 21 |
|17. $ y = e^e $ | 2 |
|18. $ y = \frac{x^2}{e^{2x}} $ | 19 |
|19. $ y = \ln(x)\sin(x) $ | 18 |
|20. $ y = \ln(\sin x) $ | 22 |
|21. $ y = \tan^5(\ln(3x^2+3)+8x) $ | 20 |
|22. $ y = x^\pi $ | 1 |
|23. $ y = \pi^x $ | 4 |
|24. $ y = \pi^\pi $ | 2 |
|25. $ y = x^x $ | 24 |
|26. $ y = \frac{\sin(3x)}{\sqrt{x^2 - 5}} $ | 19 |
|27. $ y = \cot^{-1}\left(\frac{1}{1+x^2}\right) $ | 16 |
|28. $ y = 3x^2 - \cos^{-1}x $ | 26 |

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📝 Summary Notes:

- #1: Power rule
- #2: Constant rule
- #10: Derivative of $ \cot x $
- #13: $ \frac{d}{dx}[\sin^{-1} u] $
- #15: $ \frac{d}{dx}[\tan^{-1} u] $
- #16: $ \frac{d}{dx}[\cot^{-1} u] $
- #17: Chain Rule
- #18: Product Rule
- #19: Quotient Rule
- #20: Power rule with chain
- #21: $ \frac{d}{dx}[e^u] = e^u u' $
- #22: $ \frac{d}{dx}[\ln u] = \frac{1}{u} u' $
- #24: $ \frac{d}{dx}[u^v] $ — logarithmic differentiation
- #26: Sum/Difference Rule

✔ These matches are based on standard derivative formula lists. If your formula sheet has different numbering, adjust accordingly, but this should be very close.

Let me know if you want the actual derivative solutions too!