Calaméo - Power Rule Practice Worksheet - Free Printable

Let's solve each problem on the Derivative Power Rule Practice Worksheet using the power rule for derivatives:

🔹 Power Rule:

> If $ y = x^n $, then $ \frac{dy}{dx} = nx^{n-1} $

Also remember:
- Constants become zero when differentiated.
- Use the chain rule or quotient rule if needed (but most here are simple power functions).
- Convert roots to exponents: $ \sqrt{x} = x^{1/2} $, $ \sqrt[3]{x} = x^{1/3} $, etc.

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1) $ y = x^4 $

Apply the power rule:
$$
\frac{dy}{dx} = 4x^{4-1} = 4x^3
$$

✔ Answer: $ 4x^3 $

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2) $ y = \frac{1}{x^3} $

Rewrite as a negative exponent:
$$
y = x^{-3}
$$
Now apply power rule:
$$
\frac{dy}{dx} = -3x^{-4} = -\frac{3}{x^4}
$$

✔ Answer: $ -\frac{3}{x^4} $

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3) $ y = \frac{1}{x^2} $

Rewrite:
$$
y = x^{-2}
$$
Differentiate:
$$
\frac{dy}{dx} = -2x^{-3} = -\frac{2}{x^3}
$$

✔ Answer: $ -\frac{2}{x^3} $

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4) $ f(x) = \sqrt[3]{x} $

Convert cube root to exponent:
$$
f(x) = x^{1/3}
$$
Differentiate:
$$
f'(x) = \frac{1}{3}x^{(1/3)-1} = \frac{1}{3}x^{-2/3} = \frac{1}{3x^{2/3}}
$$

✔ Answer: $ \frac{1}{3x^{2/3}} $

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5) $ f(t) = -2t^2 + 3t - 6 $

Differentiate term by term:
- Derivative of $ -2t^2 $: $ -4t $
- Derivative of $ 3t $: $ 3 $
- Derivative of $ -6 $: $ 0 $

So:
$$
f'(t) = -4t + 3
$$

✔ Answer: $ -4t + 3 $

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6) $ y = \frac{5}{2x^3} $

Rewrite:
$$
y = \frac{5}{2} \cdot x^{-3}
$$
Differentiate:
$$
\frac{dy}{dx} = \frac{5}{2} \cdot (-3)x^{-4} = -\frac{15}{2}x^{-4} = -\frac{15}{2x^4}
$$

✔ Answer: $ -\frac{15}{2x^4} $

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7) $ y = \frac{3}{2x^5} $

Rewrite:
$$
y = \frac{3}{2} \cdot x^{-5}
$$
Differentiate:
$$
\frac{dy}{dx} = \frac{3}{2} \cdot (-5)x^{-6} = -\frac{15}{2}x^{-6} = -\frac{15}{2x^6}
$$

✔ Answer: $ -\frac{15}{2x^6} $

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8) $ y = \frac{6}{(3x)^2} $

First simplify the denominator:
$$
(3x)^2 = 9x^2
\Rightarrow y = \frac{6}{9x^2} = \frac{2}{3x^2}
$$

Now write as:
$$
y = \frac{2}{3} \cdot x^{-2}
$$
Differentiate:
$$
\frac{dy}{dx} = \frac{2}{3} \cdot (-2)x^{-3} = -\frac{4}{3}x^{-3} = -\frac{4}{3x^3}
$$

✔ Answer: $ -\frac{4}{3x^3} $

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✔ Final Answers Summary:

| Problem | Derivative |
|--------|------------|
| 1) $ y = x^4 $ | $ 4x^3 $ |
| 2) $ y = \frac{1}{x^3} $ | $ -\frac{3}{x^4} $ |
| 3) $ y = \frac{1}{x^2} $ | $ -\frac{2}{x^3} $ |
| 4) $ f(x) = \sqrt[3]{x} $ | $ \frac{1}{3x^{2/3}} $ |
| 5) $ f(t) = -2t^2 + 3t - 6 $ | $ -4t + 3 $ |
| 6) $ y = \frac{5}{2x^3} $ | $ -\frac{15}{2x^4} $ |
| 7) $ y = \frac{3}{2x^5} $ | $ -\frac{15}{2x^6} $ |
| 8) $ y = \frac{6}{(3x)^2} $ | $ -\frac{4}{3x^3} $ |

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Let me know if you'd like these steps explained further or visualized!