Comprehensive calculus worksheet designed to help students practice finding derivatives using the power rule.

Let's solve each of these derivative problems using the Power Rule.

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🔷 Power Rule Recap:

For a function $ y = ax^n $, the derivative is:

$$
\frac{dy}{dx} = a \cdot n \cdot x^{n-1}
$$

We apply this rule term by term for polynomials or expressions involving powers of $ x $. Remember:
- Constants multiply the derivative.
- Negative and fractional exponents are handled the same way.
- Roots can be rewritten as fractional exponents.

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Now let’s go through each problem one by one.

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1. $ y = 10x^3 $

Apply power rule:
$$
\frac{dy}{dx} = 10 \cdot 3 \cdot x^{3-1} = 30x^2
$$

✔ Answer: $ \boxed{30x^2} $

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2. $ y = \frac{1}{2}x^{-2} $

$$
\frac{dy}{dx} = \frac{1}{2} \cdot (-2) \cdot x^{-2-1} = -1 \cdot x^{-3} = -x^{-3}
$$

✔ Answer: $ \boxed{-x^{-3}} $ or $ \boxed{-\frac{1}{x^3}} $

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3. $ y = \frac{1}{2\sqrt{x}} $

First rewrite $ \sqrt{x} = x^{1/2} $, so:
$$
y = \frac{1}{2} x^{-1/2}
$$

Now differentiate:
$$
\frac{dy}{dx} = \frac{1}{2} \cdot \left(-\frac{1}{2}\right) x^{-1/2 - 1} = -\frac{1}{4} x^{-3/2}
$$

✔ Answer: $ \boxed{-\frac{1}{4}x^{-3/2}} $ or $ \boxed{-\frac{1}{4x^{3/2}}} $

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4. $ y = 3x^{-1/15} $

$$
\frac{dy}{dx} = 3 \cdot \left(-\frac{1}{15}\right) x^{-1/15 - 1} = -\frac{1}{5} x^{-16/15}
$$

✔ Answer: $ \boxed{-\frac{1}{5}x^{-16/15}} $

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5. $ y = 8x^6 + 2x^{17} $

Differentiate term by term:
- $ \frac{d}{dx}(8x^6) = 8 \cdot 6 x^5 = 48x^5 $
- $ \frac{d}{dx}(2x^{17}) = 2 \cdot 17 x^{16} = 34x^{16} $

✔ Answer: $ \boxed{48x^5 + 34x^{16}} $

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6. $ y = \sqrt[5]{x} $

Rewrite: $ y = x^{1/5} $

$$
\frac{dy}{dx} = \frac{1}{5} x^{1/5 - 1} = \frac{1}{5} x^{-4/5}
$$

✔ Answer: $ \boxed{\frac{1}{5}x^{-4/5}} $ or $ \boxed{\frac{1}{5x^{4/5}}} $

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7. $ y = x^{1/31} + x^{-1/7} $

Differentiate each term:
- $ \frac{d}{dx}(x^{1/31}) = \frac{1}{31} x^{1/31 - 1} = \frac{1}{31} x^{-30/31} $
- $ \frac{d}{dx}(x^{-1/7}) = -\frac{1}{7} x^{-1/7 - 1} = -\frac{1}{7} x^{-8/7} $

✔ Answer: $ \boxed{\frac{1}{31}x^{-30/31} - \frac{1}{7}x^{-8/7}} $

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8. $ y = 2x^{12} + 6x^7 + x^4 $

Differentiate each term:
- $ 2 \cdot 12 x^{11} = 24x^{11} $
- $ 6 \cdot 7 x^6 = 42x^6 $
- $ 1 \cdot 4 x^3 = 4x^3 $

✔ Answer: $ \boxed{24x^{11} + 42x^6 + 4x^3} $

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9. $ y = \frac{5}{3}x^3 - \frac{7}{6}x^6 + \frac{6}{4}x^8 $

Simplify $ \frac{6}{4} = \frac{3}{2} $, so:
$$
y = \frac{5}{3}x^3 - \frac{7}{6}x^6 + \frac{3}{2}x^8
$$

Now differentiate:
- $ \frac{5}{3} \cdot 3 x^2 = 5x^2 $
- $ -\frac{7}{6} \cdot 6 x^5 = -7x^5 $
- $ \frac{3}{2} \cdot 8 x^7 = 12x^7 $

✔ Answer: $ \boxed{5x^2 - 7x^5 + 12x^7} $

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10. $ y = \frac{1}{2}x^{3/2} - \frac{22}{7}x^{-5/2} + x^{3/7} $

Differentiate each term:
1. $ \frac{1}{2} \cdot \frac{3}{2} x^{3/2 - 1} = \frac{3}{4} x^{1/2} $
2. $ -\frac{22}{7} \cdot \left(-\frac{5}{2}\right) x^{-5/2 - 1} = \frac{110}{14} x^{-7/2} = \frac{55}{7} x^{-7/2} $
3. $ 1 \cdot \frac{3}{7} x^{3/7 - 1} = \frac{3}{7} x^{-4/7} $

✔ Answer: $ \boxed{\frac{3}{4}x^{1/2} + \frac{55}{7}x^{-7/2} + \frac{3}{7}x^{-4/7}} $

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✔ Final Answers Summary:

| Problem | Derivative |
|--------|------------|
| 1. $ y = 10x^3 $ | $ 30x^2 $ |
| 2. $ y = \frac{1}{2}x^{-2} $ | $ -x^{-3} $ |
| 3. $ y = \frac{1}{2\sqrt{x}} $ | $ -\frac{1}{4}x^{-3/2} $ |
| 4. $ y = 3x^{-1/15} $ | $ -\frac{1}{5}x^{-16/15} $ |
| 5. $ y = 8x^6 + 2x^{17} $ | $ 48x^5 + 34x^{16} $ |
| 6. $ y = \sqrt[5]{x} $ | $ \frac{1}{5}x^{-4/5} $ |
| 7. $ y = x^{1/31} + x^{-1/7} $ | $ \frac{1}{31}x^{-30/31} - \frac{1}{7}x^{-8/7} $ |
| 8. $ y = 2x^{12} + 6x^7 + x^4 $ | $ 24x^{11} + 42x^6 + 4x^3 $ |
| 9. $ y = \frac{5}{3}x^3 - \frac{7}{6}x^6 + \frac{6}{4}x^8 $ | $ 5x^2 - 7x^5 + 12x^7 $ |
| 10. $ y = \frac{1}{2}x^{3/2} - \frac{22}{7}x^{-5/2} + x^{3/7} $ | $ \frac{3}{4}x^{1/2} + \frac{55}{7}x^{-7/2} + \frac{3}{7}x^{-4/7} $ |

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