Let’s solve each of the first 5 problems step by step, using basic derivative rules.
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1. Find the derivative of: `f(x) = 7`
✔ Answer: D. 0
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Explanation:
The derivative of a constant function is always
0.
Since `f(x) = 7` doesn’t change as `x` changes, its rate of change (derivative) is zero.
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2. Find the derivative of: `f(x) = -4x`
✔ Answer: D. -4
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Explanation:
This is a linear function. The derivative of `ax` (where `a` is constant) is just `a`.
So, derivative of `-4x` is `-4`.
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3. Find the derivative of: `f(x) = x⁴`
✔ Answer: B. 4x³
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Explanation:
Use the
Power Rule:
If `f(x) = xⁿ`, then `f’(x) = n·xⁿ⁻¹`
Here, `n = 4`, so `f’(x) = 4x³`
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4. Find the derivative of: `f(x) = 1 - x²`
✔ Answer: A. -2x
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Explanation:
Differentiate term by term.
- Derivative of constant `1` is `0`
- Derivative of `-x²` is `-2x` (using power rule: `d/dx[x²] = 2x`, so `d/dx[-x²] = -2x`)
→ Total: `0 - 2x = -2x`
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5. Find the derivative of: `f(x) = 1/x²`
*(Hint: rewrite with negative exponent)*
✔ Answer: C. f’(x) = -2x⁻³
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Explanation:
Rewrite: `f(x) = x⁻²`
Now apply Power Rule:
`f’(x) = -2x⁻³`
Which is the same as `-2/x³`
Option C matches this exactly.
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✔ Final Answers:
1.
D. 0
2.
D. -4
3.
B. 4x³
4.
A. -2x
5.
C. f’(x) = -2x⁻³
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Let me know if you’d like help with question #6 or the rest of the 17 questions!
Parent Tip: Review the logic above to help your child master the concept of derivative practice worksheet.
