Practice finding derivatives of polynomial and radical functions with this comprehensive power rule worksheet.

Let’s solve each problem one by one using the Power Rule for derivatives.

The Power Rule says:
If you have a function like \( y = x^n \), then its derivative is \( \frac{dy}{dx} = n \cdot x^{n-1} \).
Also, if there’s a number in front (like \( 5x^3 \)), you just multiply that number by the exponent and reduce the exponent by 1.

We’ll apply this to every problem.

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Problem 1: \( y = 10x^3 \)

Multiply coefficient (10) by exponent (3):
\( 10 \cdot 3 = 30 \)
Reduce exponent by 1: \( 3 - 1 = 2 \)
→ Derivative: \( 30x^2 \)

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Problem 2: \( y = \frac{1}{2}x^{-2} \)

Multiply coefficient (\( \frac{1}{2} \)) by exponent (-2):
\( \frac{1}{2} \cdot (-2) = -1 \)
Reduce exponent by 1: \( -2 - 1 = -3 \)
→ Derivative: \( -x^{-3} \) or \( -\frac{1}{x^3} \)

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Problem 3: \( y = \frac{1}{2\sqrt{x}} \)

First, rewrite square root as exponent:
\( \sqrt{x} = x^{1/2} \), so \( \frac{1}{2\sqrt{x}} = \frac{1}{2} x^{-1/2} \)

Now apply power rule:
Coefficient: \( \frac{1}{2} \), exponent: \( -\frac{1}{2} \)
Multiply: \( \frac{1}{2} \cdot (-\frac{1}{2}) = -\frac{1}{4} \)
New exponent: \( -\frac{1}{2} - 1 = -\frac{3}{2} \)
→ Derivative: \( -\frac{1}{4} x^{-3/2} \) or \( -\frac{1}{4x^{3/2}} \) or \( -\frac{1}{4x\sqrt{x}} \)

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Problem 4: \( y = 3x^{-1/15} \)

Multiply coefficient (3) by exponent (\( -\frac{1}{15} \)):
\( 3 \cdot (-\frac{1}{15}) = -\frac{3}{15} = -\frac{1}{5} \)
New exponent: \( -\frac{1}{15} - 1 = -\frac{16}{15} \)
→ Derivative: \( -\frac{1}{5} x^{-16/15} \)

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Problem 5: \( y = 8x^6 + 2x^{17} \)

Take derivative of each term separately.

First term: \( 8x^6 \) → \( 8 \cdot 6 = 48 \), exponent becomes 5 → \( 48x^5 \)
Second term: \( 2x^{17} \) → \( 2 \cdot 17 = 34 \), exponent becomes 16 → \( 34x^{16} \)
→ Derivative: \( 48x^5 + 34x^{16} \)

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Problem 6: \( y = \sqrt[5]{x} \)

Rewrite as exponent: \( x^{1/5} \)

Apply power rule:
Exponent is \( \frac{1}{5} \), so derivative is \( \frac{1}{5} x^{(1/5)-1} = \frac{1}{5} x^{-4/5} \)
→ Derivative: \( \frac{1}{5} x^{-4/5} \) or \( \frac{1}{5x^{4/5}} \)

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Problem 7: \( y = x^{1/31} + x^{-1/7} \)

First term: \( x^{1/31} \) → derivative: \( \frac{1}{31} x^{(1/31)-1} = \frac{1}{31} x^{-30/31} \)
Second term: \( x^{-1/7} \) → derivative: \( -\frac{1}{7} x^{(-1/7)-1} = -\frac{1}{7} x^{-8/7} \)
→ Derivative: \( \frac{1}{31} x^{-30/31} - \frac{1}{7} x^{-8/7} \)

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Problem 8: \( y = 2x^{12} + 6x^7 + x^4 \)

Derivative term by term:

- \( 2x^{12} \) → \( 2 \cdot 12 = 24 \), exponent 11 → \( 24x^{11} \)
- \( 6x^7 \) → \( 6 \cdot 7 = 42 \), exponent 6 → \( 42x^6 \)
- \( x^4 \) → \( 1 \cdot 4 = 4 \), exponent 3 → \( 4x^3 \)

→ Derivative: \( 24x^{11} + 42x^6 + 4x^3 \)

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Problem 9: \( y = \frac{5}{3}x^3 - \frac{7}{6}x^6 + \frac{6}{4}x^8 \)

Simplify fractions first: \( \frac{6}{4} = \frac{3}{2} \), so we have:

\( y = \frac{5}{3}x^3 - \frac{7}{6}x^6 + \frac{3}{2}x^8 \)

Now take derivatives:

- First term: \( \frac{5}{3} \cdot 3 = 5 \), exponent 2 → \( 5x^2 \)
- Second term: \( -\frac{7}{6} \cdot 6 = -7 \), exponent 5 → \( -7x^5 \)
- Third term: \( \frac{3}{2} \cdot 8 = 12 \), exponent 7 → \( 12x^7 \)

→ Derivative: \( 5x^2 - 7x^5 + 12x^7 \)

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Problem 10: \( y = \frac{1}{2}x^{3/2} - \frac{22}{7}x^{-5/2} + x^{3/7} \)

Term by term:

1. \( \frac{1}{2}x^{3/2} \) → \( \frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4} \), new exponent: \( \frac{3}{2} - 1 = \frac{1}{2} \) → \( \frac{3}{4}x^{1/2} \)

2. \( -\frac{22}{7}x^{-5/2} \) → \( -\frac{22}{7} \cdot (-\frac{5}{2}) = \frac{110}{14} = \frac{55}{7} \), new exponent: \( -\frac{5}{2} - 1 = -\frac{7}{2} \) → \( \frac{55}{7}x^{-7/2} \)

3. \( x^{3/7} \) → \( 1 \cdot \frac{3}{7} = \frac{3}{7} \), new exponent: \( \frac{3}{7} - 1 = -\frac{4}{7} \) → \( \frac{3}{7}x^{-4/7} \)

→ Derivative: \( \frac{3}{4}x^{1/2} + \frac{55}{7}x^{-7/2} + \frac{3}{7}x^{-4/7} \)

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Final Answer:

1. \( 30x^2 \)
2. \( -x^{-3} \) or \( -\frac{1}{x^3} \)
3. \( -\frac{1}{4}x^{-3/2} \) or \( -\frac{1}{4x^{3/2}} \)
4. \( -\frac{1}{5}x^{-16/15} \)
5. \( 48x^5 + 34x^{16} \)
6. \( \frac{1}{5}x^{-4/5} \) or \( \frac{1}{5x^{4/5}} \)
7. \( \frac{1}{31}x^{-30/31} - \frac{1}{7}x^{-8/7} \)
8. \( 24x^{11} + 42x^6 + 4x^3 \)
9. \( 5x^2 - 7x^5 + 12x^7 \)
10. \( \frac{3}{4}x^{1/2} + \frac{55}{7}x^{-7/2} + \frac{3}{7}x^{-4/7} \)