Let's solve each problem step by step using the
Power Rule for derivatives.
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🔷 Power Rule:
If $ f(x) = ax^n $, then
$$
f'(x) = a \cdot n \cdot x^{n-1}
$$
We'll apply this rule to each function after rewriting it as a power function (if needed).
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✔ Problem 1:
Given: $ f(x) = \frac{x^7}{3} $
#### Step 1: Rewrite as a multiple of a power function:
$$
f(x) = \frac{1}{3}x^7
$$
#### Step 2: Apply Power Rule:
$$
f'(x) = \frac{1}{3} \cdot 7 \cdot x^{6} = \frac{7}{3}x^6
$$
✔ Answer:
| $ f(x) $ | $ f(x) $ expressed as... | $ f'(x) $ |
|-----------|-----------------------------|------------|
| $ \frac{x^7}{3} $ | $ \frac{1}{3}x^7 $ | $ \frac{7}{3}x^6 $ |
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✔ Problem 2:
Given: $ f(x) = \frac{1}{x^7} $
#### Step 1: Rewrite using negative exponent:
$$
f(x) = x^{-7}
$$
#### Step 2: Apply Power Rule:
$$
f'(x) = -7x^{-8} = -\frac{7}{x^8}
$$
✔ Answer:
| $ f(x) $ | $ f(x) $ expressed as... | $ f'(x) $ |
|-----------|-----------------------------|------------|
| $ \frac{1}{x^7} $ | $ x^{-7} $ | $ -7x^{-8} $ or $ -\frac{7}{x^8} $ |
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✔ Problem 3:
Given: $ f(x) = \sqrt[4]{x} $
#### Step 1: Rewrite using fractional exponent:
$$
f(x) = x^{1/4}
$$
#### Step 2: Apply Power Rule:
$$
f'(x) = \frac{1}{4}x^{-3/4}
$$
✔ Answer:
| $ f(x) $ | $ f(x) $ expressed as... | $ f'(x) $ |
|-----------|-----------------------------|------------|
| $ \sqrt[4]{x} $ | $ x^{1/4} $ | $ \frac{1}{4}x^{-3/4} $ |
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✔ Problem 4:
Given: $ f(x) = -\frac{3}{x} $
#### Step 1: Rewrite:
$$
f(x) = -3x^{-1}
$$
#### Step 2: Apply Power Rule:
$$
f'(x) = -3 \cdot (-1) \cdot x^{-2} = 3x^{-2} = \frac{3}{x^2}
$$
✔ Answer:
| $ f(x) $ | $ f(x) $ expressed as... | $ f'(x) $ |
|-----------|-----------------------------|------------|
| $ -\frac{3}{x} $ | $ -3x^{-1} $ | $ 3x^{-2} $ or $ \frac{3}{x^2} $ |
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✔ Problem 5:
Given: $ f(x) = 6\sqrt[4]{x^7} $
#### Step 1: Rewrite radical as exponent:
$$
\sqrt[4]{x^7} = x^{7/4}
\Rightarrow f(x) = 6x^{7/4}
$$
#### Step 2: Apply Power Rule:
$$
f'(x) = 6 \cdot \frac{7}{4} \cdot x^{(7/4)-1} = \frac{42}{4}x^{3/4} = \frac{21}{2}x^{3/4}
$$
✔ Answer:
| $ f(x) $ | $ f(x) $ expressed as... | $ f'(x) $ |
|-----------|-----------------------------|------------|
| $ 6\sqrt[4]{x^7} $ | $ 6x^{7/4} $ | $ \frac{21}{2}x^{3/4} $ |
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🟩 Final Table:
| $ f(x) $ | $ f(x) $ expressed as a power function | $ f'(x) $ |
|----------|----------------------------------------|------------|
| Example: $ \frac{5x^6}{4} $ | $ \frac{5}{4}x^6 $ | $ \frac{15}{2}x^5 $ |
| 1. $ \frac{x^7}{3} $ | $ \frac{1}{3}x^7 $ | $ \frac{7}{3}x^6 $ |
| 2. $ \frac{1}{x^7} $ | $ x^{-7} $ | $ -7x^{-8} $ |
| 3. $ \sqrt[4]{x} $ | $ x^{1/4} $ | $ \frac{1}{4}x^{-3/4} $ |
| 4. $ -\frac{3}{x} $ | $ -3x^{-1} $ | $ 3x^{-2} $ |
| 5. $ 6\sqrt[4]{x^7} $ | $ 6x^{7/4} $ | $ \frac{21}{2}x^{3/4} $ |
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Let me know if you'd like these answers in simplified fractional form or with radicals!
Parent Tip: Review the logic above to help your child master the concept of derivatives power rule worksheet.
