Step by Step
1
The rule itself
For any term xⁿ, the derivative is n·xⁿ⁻¹ — bring the exponent down in front as a multiplier, then subtract 1 from the exponent.
2
Special cases
The derivative of a plain x (which is x¹) is just 1. The derivative of any constant by itself is always 0, since a constant never changes as x changes.
3
Works term by term
For a polynomial with multiple terms, apply the power rule separately to each term, then add the results together — you never need to touch the terms in relation to each other.
4
Coefficients just carry through
If a term already has a coefficient, like 5x³, multiply that coefficient by the new exponent: d/dx[5x³] = 5·3·x² = 15x².
Applied Walkthrough
1
Differentiate f(x) = x⁵ + 3x² − 7x + 4. Handle each term separately using the power rule.
2
d/dx[x⁵] = 5x⁴. d/dx[3x²] = 3·2·x¹ = 6x.
3
d/dx[−7x] = −7 (since x is x¹, and the rule gives 1·(−7)·x⁰ = −7). d/dx[4] = 0, since 4 is a constant.
4
Adding these together: f'(x) = 5x⁴ + 6x − 7.
Exam Application
Exams test whether you can apply the power rule correctly to every term in a polynomial, including handling coefficients, plain x terms, and constants correctly.
⚠ Common Trap
The most common trap is forgetting that the derivative of a lone constant is 0 — not the constant itself. Another frequent mistake is forgetting to multiply the existing coefficient by the new exponent.
✓ Quick Self-Check
1. What is the power rule formula?
d/dx[xⁿ] = n·xⁿ⁻¹.
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2. What is the derivative of a constant?
0.
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3. Differentiate f(x) = x⁵ + 3x² − 7x + 4.
f'(x) = 5x⁴ + 6x − 7.
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4. What is the derivative of x by itself?
1.
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5. How do you handle a term that already has a coefficient, like 5x³?
Multiply the existing coefficient by the new exponent: d/dx[5x³] = 15x².
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