AP Calculus AB/BC Unit 2 Cheat Sheet
A one-page visual summary of the Definition of the Derivative — every rule, derivative, and exam trap you need, on a single screen.
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The basics
What it covers: Defining the derivative as a limit, connecting it to tangent-line slope and continuity, and the fundamental differentiation rules.
Exam weight: About 10–12% of the AB exam and 4–7% of the BC exam.
The big question: How do we measure an instantaneous rate of change, and how do we differentiate efficiently?
Mathematical practices: Implementing Mathematical Processes, Connecting Representations, Justification, and Communication & Notation.
Key topics at a glance
Rates of Change
Average rate = slope of the secant = [f(b) − f(a)]/(b − a). Instantaneous rate = the derivative = limit of average rates.
The Derivative
f′(x) = lim(h→0) [f(x+h) − f(x)]/h. Point form: f′(a) = lim(x→a) [f(x) − f(a)]/(x − a). Notation: f′, y′, dy/dx.
Derivative as Slope
f′(a) is the slope of the tangent line at (a, f(a)). Tangent line: y − f(a) = f′(a)(x − a).
Differentiability & Continuity
Differentiable ⇒ continuous (not the reverse). Derivatives fail at corners, cusps, vertical tangents, and discontinuities.
Basic Rules
Power: d/dx[xⁿ] = n·xⁿ⁻¹. Constant: d/dx[c] = 0. Constant multiple: (c·f)′ = c·f′. Sum/Diff: (f ± g)′ = f′ ± g′.
Key Derivatives
sin x → cos x; cos x → −sin x; eᵇ → eᵇ; ln x → 1/x.
Product & Quotient
Product: (fg)′ = f′g + fg′. Quotient: (f/g)′ = (f′g − fg′)/g².
Trig Derivatives
tan x → sec²x; sec x → sec x tan x; cot x → −csc²x; csc x → −csc x cot x.
The key terms and rules you must know
- Average rate of change — [f(b) − f(a)]/(b − a), the secant slope.
- Instantaneous rate of change — the derivative; limit of average rates.
- Derivative (limit definition) — f′(x) = lim(h→0) [f(x+h) − f(x)]/h.
- Tangent line — y − f(a) = f′(a)(x − a).
- Differentiability ⇒ continuity — but not the reverse.
- Power Rule — d/dx[xⁿ] = n·xⁿ⁻¹.
- Constant / constant-multiple / sum rules — the linearity of derivatives.
- sin x, cos x — derivatives cos x and −sin x.
- eᵇ, ln x — derivatives eᵇ and 1/x.
- Product Rule — (fg)′ = f′g + fg′.
- Quotient Rule — (f/g)′ = (f′g − fg′)/g².
- Trig derivatives — tan′ = sec², sec′ = sec tan, cot′ = −csc², csc′ = −csc cot.
Key themes to remember
- The derivative is a limit. Every rule is a shortcut for the difference-quotient limit.
- Derivative = slope of the tangent. This links algebra to geometry throughout calculus.
- Smooth means differentiable. Corners, cusps, and vertical tangents break differentiability.
- Master the rule table. Power, product, quotient, and the key function derivatives are used constantly.
- Watch the signs. cos x → −sin x, and the quotient rule subtracts in a specific order.
Common exam traps
- Continuity does not imply differentiability. |x| is continuous at 0 but has no derivative there.
- The derivative of cos x is −sin x — don't forget the negative sign.
- Quotient rule order matters: (f′g − fg′)/g², not (fg′ − f′g). Low d-high minus high d-low.
- The product rule is not f′g′. It is f′g + fg′.
- Power rule needs a constant exponent. It does not apply to eᵇ (use d/dx[eᵇ] = eᵇ).
- An infinite/undefined slope means no derivative. A vertical tangent is not differentiable.
- Rewrite before differentiating when possible. A radical or fraction is often easier as a power of x.