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10 multiple-choice questions and 18 flashcards on Differentiation: Definition and Basic Rules, about 7% of the High School Calculus bank. Every one carries a written rationale.
Differentiation: Definition and Basic Rules is one of 8 chapters in CoStudy's High School Calculus bank, and it holds 10 of the bank's 150 multiple-choice questions — roughly 7% of the total. That proportion is not arbitrary: chapters follow the certifying body's published exam outline, and the number of questions in each is set by that domain's published weight, so the share of your practice time this chapter takes matches the share of the real exam it accounts for.
Studying by chapter is worth doing once you have a diagnostic score. A single overall percentage tells you whether you are close; it does not tell you which domain is dragging. Working a weak chapter in isolation, and re-testing it in isolation, is the fastest way to move a score that has stalled — and it is why the mock exams in CoStudy report by domain rather than as one number.
4 questions drawn from this chapter, with the full rationale shown — the controlling principle behind the right answer, and why each wrong option tempts and fails.
d/dx [x³] =
Answer: D — 3x²
D) Power rule: d/dx[xⁿ] = nxⁿ⁻¹. B) Dropped the coefficient. C) Computed an antiderivative instead. A) Reduced exponent without applying the rule.
What is the SECOND derivative used for?
Answer: E — Concavity and acceleration; f'' > 0 means concave up; f'' < 0 means concave down; inflection points where concavity changes
f'(x): slope/velocity. f''(x): rate of change of slope, concavity, acceleration. Inflection points: f''(x) = 0 and concavity changes. Together f' and f'' classify critical points (Second Derivative Test).
What does the FUNDAMENTAL THEOREM OF CALCULUS state?
Answer: B — Differentiation and integration are inverse operations: ∫ₐᵇ f'(x) dx = f(b) - f(a) (Part 2); d/dx[∫ₐˣ f(t) dt] = f(x) (Part 1)
FTC connects two main branches of calculus. Part 1: derivative of integral function = integrand. Part 2: definite integral = difference of antiderivatives at bounds. Foundation of integral calculus.
Find the SLOPE of the tangent line to y = x² at x = 3.
Answer: B — 6 (slope = y'(3) = 2·3 = 6)
Slope of tangent line = derivative at that point. y' = 2x. At x=3: y'(3) = 6. Tangent line: y - 9 = 6(x - 3). Point-slope form using point (3, 9) and slope 6.
4 cards from the 18 in this chapter.
Find d/dx[(2x + 1)³] using chain rule.
3(2x + 1)² · 2 = 6(2x + 1)².
What is the power rule?
d/dx[xⁿ] = n·xⁿ⁻¹.
What is the constant multiple rule?
d/dx[c·f(x)] = c·f'(x).
Find d/dx[x¹/²].
(1/2)x⁻¹/² = 1/(2√x).
These are a sample. The full Differentiation: Definition and Basic Rules chapter runs 28 items with per-chapter progress tracking, on the web and in the iOS app.
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