AP Calculus: Power Rule, Constant Multiple Rule, Sum/Difference Rule

Power Rule, Constant Multiple Rule, Sum/Difference Rule

Concept Summary

• Power Rule: For any real number n, the derivative of xⁿ is nxⁿ⁻¹; the foundational shortcut for differentiating polynomial terms.
• Constant Multiple Rule: The derivative of c·f(x) is c·f'(x); constants factor out of differentiation, preserving linearity.
• Sum/Difference Rule: The derivative of f(x) ± g(x) is f'(x) ± g'(x); differentiation distributes over addition/subtraction.
• Linearity of Differentiation: The combination of the Constant Multiple and Sum/Difference Rules; enables term-by-term differentiation of polynomials.
• Domain Consistency: All rules apply only where the original function is differentiable; discontinuities or sharp corners invalidate the rules.

Core Questions

WHAT (definitional)

Q: What is the Power Rule? A: The Power Rule states that if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹ for any real number n.
⚠️ Trap/Clarification: The exponent n must be a constant; xˣ or x^sin(x) do not follow the Power Rule.

Q: What is the Constant Multiple Rule? A: The Constant Multiple Rule says the derivative of c·f(x) is c·f'(x), where c is a constant.
⚠️ Trap/Clarification: c must be a constant (e.g., 5, π); if c is a function (e.g., x·f(x)), the Product Rule applies instead.

WHY (causal/explanatory)

Q: Why does the Power Rule work? A: The Power Rule is derived from the limit definition of the derivative, using binomial expansion for n as a positive integer and extended via limits/algebra to all real n.
⚠️ Trap/Clarification: It’s not just "bring the exponent down and subtract 1"; this is a mnemonic, not the proof.

Q: Why is the Sum/Difference Rule important? A: It allows differentiation of polynomials (and other sums/differences) term-by-term, simplifying complex expressions into manageable parts.
⚠️ Trap/Clarification: It does not apply to products/quotients (e.g., f(x)·g(x) requires the Product Rule).

HOW (process/application)

Q: How do you apply the Power Rule? A: Multiply the term by its exponent, then subtract 1 from the exponent: d/dx [xⁿ] = nxⁿ⁻¹.
⚠️ Trap/Clarification: Forgetting to subtract 1 from the exponent (e.g., d/dx [x³] = 3x³ is wrong; it’s 3x²).

Q: How do you differentiate 5x⁴ - 3x² + 7? A: Apply the Power Rule to each term, then use the Sum/Difference and Constant Multiple Rules: 20x³ - 6x + 0.
⚠️ Trap/Clarification: The derivative of a constant (e.g., 7) is 0, not 7x⁰ or 1.

CAN (conditions/possibilities)

Q: Can the Power Rule be used for √x or 1/x? A: Yes: rewrite √x as x^(1/2) (derivative: (1/2)x^(-1/2)) and 1/x as x^(-1) (derivative: -x^(-2)).
⚠️ Trap/Clarification: Negative/ fractional exponents are valid; don’t revert to the limit definition for these cases.

Q: Under what conditions does the Sum Rule fail? A: The Sum Rule fails if either f(x) or g(x) is not differentiable at the point of interest (e.g., |x| + x² at x = 0).
⚠️ Trap/Clarification: Differentiability is not guaranteed just because the sum is continuous.

Quick Facts & Traps

• Fact: The Power Rule works for all real exponents (e.g., x^π → πx^(π-1)).
• Trap: "d/dx [c] = c" → Reality: The derivative of any constant c is 0.
• Fact: The Sum Rule does not require f and g to be differentiable everywhere—just at the point where you’re differentiating.
• Trap: "d/dx [f(x)·g(x)] = f'(x) + g'(x)" → Reality: This is the Product Rule, not the Sum Rule.
• Fact: The Constant Multiple Rule applies to any constant, including π, e, or √2.
• Trap: "d/dx [x·f(x)] = f(x)" → Reality: This requires the Product Rule: f(x) + x·f'(x).

Rapid-Fire True/False

• Statement: The derivative of 3x² + 2x + 1 is 6x + 2.
  Answer: TRUE Why the common mistake happens: Forgetting the derivative of the constant 1 is 0 (not 1).

• Statement: The Power Rule can be used to differentiate 2ˣ.
  Answer: FALSE Why the common mistake happens: 2ˣ is an exponential function (derivative: 2ˣ ln(2)), not a power function.

• Statement: If f(x) = x³ and g(x) = x², then d/dx [f(x) + g(x)] = 3x² + 2x.
  Answer: TRUE Why the common mistake happens: Misapplying the Product Rule to a sum (e.g., writing 3x²·2x instead of 3x² + 2x).