Questions: Constant Multiple and Sum/Difference Rules

Question 1

Find d/dx[4x³ − 7x + 3].

• A. 12x² − 7x
• B. 12x² − 7
• C. 4x² − 7
• D. 12x³ − 7

Answer: (B)

Apply the rules term by term: d/dx[4x³] = 4·3x² = 12x² (constant multiple + power rule); d/dx[−7x] = −7·1 = −7 (constant multiple + power rule, since x = x¹); d/dx[3] = 0 (derivative of a constant is zero). Assembled: 12x² − 7. Option A keeps the x in the −7x term, forgetting to differentiate it (leaving −7x instead of −7). Option C forgets to multiply by 3 when applying the power rule to x³. Option D retains the cubic exponent, not reducing it by 1.

Question 2

A student differentiates h(x) = (3x²)(5x³) by writing h'(x) = (6x)(15x²) = 90x³. What error did they make?

• A. They applied the wrong exponent rule when differentiating 3x²
• B. They incorrectly assumed d/dx[f·g] = f'·g' — the derivative of a product is not the product of the derivatives
• C. They forgot to apply the constant multiple rule to factor out the 15
• D. They should have applied the chain rule here instead

Answer: (B)

The sum/difference rule (linearity of differentiation) applies to f + g, not f·g. d/dx[f·g] ≠ f'·g'. For products, the product rule gives d/dx[f·g] = f·g' + f'·g. The correct approach here is simpler: first simplify h(x) = 15x⁵, then differentiate to get h'(x) = 75x⁴. Alternatively, applying the product rule: (3x²)(15x²) + (6x)(5x³) = 45x⁴ + 30x⁴ = 75x⁴. The student's 90x³ is wrong because they distributed the derivative operation over multiplication, which is not permitted.

Question 3

The derivative of f(x) = 5x⁴ − 3x² + 2x − 9 is f'(x) = 20x³ − 6x + 2.

• T. True
• F. False

Answer: True

Apply constant multiple and sum/difference rules term by term: d/dx[5x⁴] = 5·4x³ = 20x³; d/dx[−3x²] = −3·2x = −6x; d/dx[2x] = 2·1 = 2; d/dx[−9] = 0 (constant). Result: 20x³ − 6x + 2. The constant −9 contributes zero — a common point of error is leaving it as −9 rather than 0. This example demonstrates all three aspects of linearity: constants factor out, derivatives distribute over all terms in the sum, and constant terms vanish.

Question 4

Because differentiation distributes over addition — d/dx[f + g] = f' + g' — it also distributes over multiplication: d/dx[f·g] = f'·g'.

• T. True
• F. False

Answer: False

Linearity of differentiation is a specific property of the addition operation, not a universal property of all binary operations. The product rule shows that d/dx[f·g] = f·g' + f'·g — cross terms appear, and the operation does not simply distribute. The distribution over sums follows from the algebraic property of limits (the limit of a sum is the sum of the limits), but no analogous property holds for products. Assuming linearity extends to products is one of the most common structural errors in early calculus — products need the product rule, compositions need the chain rule.

Question 5

What does it mean to say that differentiation is a 'linear operator,' and what is the most important boundary of that linearity that students must not cross?

Model answer: A linear operator satisfies two properties: scaling (d/dx[c·f] = c·f') and additivity (d/dx[f + g] = f' + g'). Together these mean you can break a sum into parts, differentiate each separately, and reassemble — the structure is fully preserved. The critical boundary is that linearity applies only to sums and differences, not to products or compositions. d/dx[f·g] ≠ f'·g' (the product rule is needed), and d/dx[f(g(x))] is not simply f'(x)·g'(x) (the chain rule is needed).

Understanding linearity as a precise property — not just vague 'distribution' — tells you exactly when the simple rules apply and when they don't. The power to differentiate polynomials term by term comes entirely from linearity; the need for the product rule and chain rule marks where linearity ends. Linearity recurs throughout mathematics (linear algebra, differential equations, integration), so recognizing its structure and limits is a durable conceptual skill, not just a calculus technique.