Calculus AB — the differentiation rules

Before the derivative can be applied to anything, it has to be computed reliably — and on the exam that means the rules coming out correctly under time pressure, without a calculator. This set covers ten of them from Units 2 and 3, one question each: the product rule, the quotient rule, a triple chain, a trigonometric derivative, an exponential whose base is not $e$, a logarithmic composite, an inverse trigonometric derivative, a derivative read off stated values, the derivative of an inverse function, and a piecewise junction.

Three questions are built around distinctions rather than manipulations. The exponential $5^{x^2}$ needs the factor $\ln (5)$; the familiar answer $5^{x^2} \cdot 2x$ is correct only for base $e$, and only because $\ln (e)=1$ — a property of that one base, not a general rule. The chain-rule question with stated values supplies both $f^{\mathrm{\prime }}(2)=6$ and $f^{\mathrm{\prime }}(4)=5$ on purpose: since $g(2)=4$, the outer derivative must be read at $4$, and a candidate who reads it at $2$ gets flawless arithmetic and the wrong answer. The piecewise question gives $x^2$ for $x\le 1$ and $3x - 2$ afterwards; the pieces meet without a gap, so the function is continuous, but the slopes arriving from either side are $2$ and $3$, so there is a corner and no derivative. One of its options — differentiable but not continuous — can be discarded before any calculation, because differentiability implies continuity and never the reverse.

Every distractor is a specific mistake, and each explanation names which one produces it. Multiplying the two derivatives of a product. Reversing the numerator of the quotient rule, or forgetting to square its denominator. Stopping after one layer of a three-layer composition. Using $\sec (x)\tan (x)$ where ${\sec }^2(x)$ belongs. Squaring the argument of $\arctan (3x)$ correctly but losing the factor of $3$ on top. Taking the logarithm of the exponent rather than of the base. The aim is that a wrong click identifies the habit to repair, rather than just marking the question red.

What this does not cover is the free-response half of the exam, where roughly half the marks live and where the working carries most of the credit. Nor does it use graphs read from a figure, which real Section I questions lean on heavily — every value here is written into the sentence instead, so each question stands on its own text.

• Apply the product and quotient rules correctly, including the order of the quotient rule's numerator and the squared denominator
• Differentiate a composition of three functions, supplying a factor for every layer
• Differentiate trigonometric and inverse trigonometric functions with the chain rule
• Differentiate an exponential whose base is not $e$, supplying the factor $\ln a$, and recognise why base $e$ is the exception rather than the pattern
• Differentiate a logarithmic composite as $u'/u$, keeping the whole argument in the denominator
• Compute the derivative of a product or composition at a point from stated values, reading the outer derivative at the inner function's output
• Find the derivative of an inverse function at a point by locating the corresponding input first
• Test a piecewise function at its junction for continuity and for differentiability separately, and explain why one implies the other in only one direction

Built against the published Course and Exam Description structure for AP Calculus AB: Unit 2 (Differentiation — Definition and Fundamental Properties: the power, product and quotient rules and the derivatives of the standard functions) and Unit 3 (Differentiation — Composite, Implicit and Inverse Functions: the chain rule, inverse functions and inverse trigonometric functions). The exam runs 3 hours 15 minutes across two sections, multiple choice and free response, each with a calculator and a no-calculator part; this set is no-calculator throughout. Every function, number and scenario here is invented for this material — nothing is reproduced from any College Board publication, released exam or scoring guideline. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the AP Calculus AB exam, and it is not an exam centre.