OMPT drill — Calculus

OMPT differentiation rules practice: lesson and 10 problems

Power, product, quotient, and chain rule. Chain rule with a function inside a function inside a power is the OMPT's favourite way to separate the prepared from the hopeful. Say the rule you are using out loud before you differentiate or integrate — it sounds silly and it works.

Tested inOMPT-A OMPT-B OMPT-D OMPT-F

OMPT differentiation rules: the lesson

What the OMPT tests in differentiation rules

The derivative $f^{'} (x)$ measures how fast $f$ is changing at each point: the slope of the graph, read as a function of position. Computing it from the definition every time would be misery, so calculus hands you a rulebook. The foundation is the power rule: $x^{n}$ differentiates to $n x^{n - 1}$ . Bring the exponent down as a multiplier, knock the exponent down by one. The Differentiation chapter sits on the OMPT-A, OMPT-B, OMPT-D and OMPT-F. On the OMPT-A and OMPT-F it covers polynomials, powers, roots, exponentials and logarithms; the OMPT-B and OMPT-D add sine and cosine.

Three combination rules then cover every function the exam can build: the product rule for things multiplied, the quotient rule for things divided, and the chain rule for things nested inside each other. Of the three, the chain rule decides grades. The OMPT's favourite construction is a function inside a function inside a power, because it forces you to apply the chain rule on purpose rather than on autopilot. Questions are phrased as "Compute $f^{'} (2)$ ", "Which expression equals $f^{'} (x)$ ", or "Find $\frac{d y}{d x}$ ". Before differentiating anything, say which rule the expression needs. Five seconds of diagnosis prevent most of the errors I mark.

Notation and answer format the OMPT expects

$f^{'} (x)$ and $\frac{d y}{d x}$ mean the same thing, and the OMPT uses both. A question asking for $\frac{d y}{d x}$ of $y = (3 x^{2} + 1)^{5}$ is a chain rule question.
A value such as $f^{'} (2)$ is typed as an exact number: $1/8$ , not 0.125, unless the question says to round.
An expression is typed with the platform's syntax: $x^{2}$ as x^2, roots as sqrt(x), $e^{x}$ as e^x or exp(x), $ln x$ as ln(x). A simplified form is easier to type without a slip.
Negative and fractional exponents may be typed as they are, so $\frac{1}{2} x^{- 1/2}$ can be entered as (1/2)x^(-1/2). Bracket the exponent.

Worked example 1: the power rule after a rewrite

Let $f (x) = 2 x^{4} - \frac{3}{x ^{2}} + 4 x$ . Compute $f^{'} (1)$ .
The power rule cannot see fractions or roots until they are exponents. Rewrite: $f (x) = 2 x^{4} - 3 x^{- 2} + 4 x^{1/2}$ .
Term by term: $2 x^{4} \to 8 x^{3}$ ; $- 3 x^{- 2} \to 6 x^{- 3}$ , because $- 3 \cdot (- 2) = 6$ and the exponent drops to $- 3$ ; $4 x^{1/2} \to 2 x^{- 1/2}$ .
So $f^{'} (x) = 8 x^{3} + \frac{6}{x ^{3}} + \frac{2}{x}$ .
At $x = 1$ : $8 + 6 + 2 = 16$ . The trap is the middle term: two negatives multiply to a plus, and the exponent goes down to $- 3$ , not up to $- 1$ .

Worked example 2: the chain rule, narrated

Let $g (x) = (2 x^{3} - 5 x)^{4}$ . Compute $g^{'} (1)$ .
Diagnose: the inside $2 x^{3} - 5 x$ is trapped inside a fourth power, the outside. Chain rule.
Outside first, leaving the inside untouched: $4 (2 x^{3} - 5 x)^{3}$ .
Times the derivative of the inside: $(2 x^{3} - 5 x)^{'} = 6 x^{2} - 5$ . So $g^{'} (x) = 4 (2 x^{3} - 5 x)^{3} (6 x^{2} - 5)$ .
At $x = 1$ the inside is $2 - 5 = - 3$ and its derivative is $6 - 5 = 1$ , so $g^{'} (1) = 4 \cdot (- 27) \cdot 1 = - 108$ . The forgotten factor $(6 x^{2} - 5)$ is the single most harvested error on OMPT calculus questions; make "times the inside" a spoken step.

The full lesson puts the whole rulebook on one card, then works the product and quotient rules, exponentials and logarithms under the chain rule, a parameter question, and the trig chain the OMPT-B and OMPT-D use. It ends with the edge cases and the eight slips I see most often.

How each OMPT variant tests differentiation rules

The chapter weights are the official ones from the omptest.org learning-assessment pages. The question counts are my estimates of how much of that chapter this one topic takes up. The 21-question papers (OMPT-D and OMPT-F) ask long multi-part questions, so a topic there usually shows up as a step inside a bigger problem.

OMPT-A
52 questions
Testedroughly 5 to 7 of 52 questions
Official chapter: Differentiation, 20% of the paper.
Numeric answers such as f'(2), or pick the correct derivative from four expressions. Power rule with roots and reciprocals, chain rule on a bracket to a power, exponentials and logarithms; no trig.
OMPT-B
64 questions
Testedroughly 5 to 8 of 64 questions
Official chapter: Differentiation, 25% of the paper.
Everything on the OMPT-A plus sine and cosine under the chain rule, and derivatives that feed directly into integration items later on the paper.
OMPT-C
30 questions
Not tested
The OMPT-C has no Differentiation chapter; its four chapters stop at linear equations and geometry, so put the hours there.
OMPT-D
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Differentiation, 20% of the paper.
Multi-part items where the derivative of a product or quotient with a parameter opens a tangent, extreme value or limit question. Trig and exponential chains are normal.
OMPT-E
40 questions
Not tested
The OMPT-E has no calculus; after functions and logarithms its chapters are statistics and probability, so drill those instead.
OMPT-F
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Differentiation, 40% of the paper.
One of six chapters on a short paper, so it carries weight: derivatives of exponential and logarithmic functions with a parameter, followed by an application part. No trig.
OMPT-G
26 questions
Not tested
The OMPT-G has no differentiation; rates of change appear in the mechanics chapter through formulas, not derivatives, so study sequences, trig and physics instead.

OMPT differentiation rules practice problems

10 original problems in the formats the OMPT uses: typed numeric answers, multiple choice, and spot-the-error. The last few are harder than the exam average on purpose. Solutions and the common mistake sit behind each problem.

Problem 1Numeric answer

Let $f (x) = x^{3} - 5 x^{2} + 2 x$ . Compute $f^{'} (2)$ .

Show the worked solution

Answer: -6

The power rule gives $f^{'} (x) = 3 x^{2} - 10 x + 2$ . Evaluating at $x = 2$ : $3 \cdot 4 - 20 + 2 = 12 - 20 + 2 = - 6$ . The curve is falling at that point.

Problem 2Multiple choice

What is the derivative of $y = x^{2} e^{x}$ ?

A $2 x e^{x}$
B $(x^{2} + 2 x) e^{x}$
C $2 x + e^{x}$
D $x^{2} e^{x - 1}$

Show the worked solution

Answer: B — $(x^{2} + 2 x) e^{x}$

This is a product, so I need the product rule: $(x^{2})^{'} e^{x} + x^{2} (e^{x})^{'} = 2 x e^{x} + x^{2} e^{x}$ . Factoring out $e^{x}$ gives $(x^{2} + 2 x) e^{x}$ .

Problem 3Numeric answer

Let $g (x) = (3 x + 1)^{5}$ . Compute $g^{'} (0)$ .

Show the worked solution

Answer: 15

The chain rule gives $g^{'} (x) = 5 (3 x + 1)^{4} \cdot 3 = 15 (3 x + 1)^{4}$ . At $x = 0$ the bracket equals 1, so $g^{'} (0) = 15$ .

Problem 4Multiple choice

What is the derivative of $y = ln (x^{2} + 1)$ ?

A $\frac{1}{x ^{2} + 1}$
B $\frac{2 x}{x ^{2} + 1}$
C $\frac{1}{2 x}$
D $2 x ln (x^{2} + 1)$

Show the worked solution

Answer: B — $\frac{2 x}{x ^{2} + 1}$

The chain rule for logarithms says the derivative is the inner derivative over the inner function: $\frac{( x ^{2} + 1 ) ^{'}}{x ^{2} + 1} = \frac{2 x}{x ^{2} + 1}$ .

Problem 5Spot the error

A student differentiates $h (x) = \frac{x}{x + 1}$ . Step 1: identify numerator $u = x$ and denominator $v = x + 1$ , with $u^{'} = 1$ and $v^{'} = 1$ . Step 2: apply the quotient rule as $h^{'} = \frac{u v ^{'} - v u ^{'}}{v ^{2}}$ . Step 3: substitute to get $h^{'} (x) = \frac{x - ( x + 1 )}{( x + 1 ) ^{2}} = \frac{- 1}{( x + 1 ) ^{2}}$ . Which step contains the error?

Show the worked solution

Answer: Step 2

The quotient rule is $h^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$ — Step 2 has the numerator terms swapped, which flips the sign of the whole result. Done correctly: $\frac{1 \cdot ( x + 1 ) - x \cdot 1}{( x + 1 ) ^{2}} = \frac{1}{( x + 1 ) ^{2}}$ . A quick sanity check: $\frac{x}{x + 1}$ is increasing for $x > - 1$ , so its derivative must be positive there.

Problem 6Numeric answer

Let $p (x) = sin x cos x$ . Compute $p^{'} (\frac{π}{3})$ as a decimal.

Show the worked solution

Answer: -0.5

By the product rule, $p^{'} (x) = cos^{2} x - sin^{2} x$ , which is exactly $cos 2 x$ . At $x = \frac{π}{3}$ : $cos \frac{2 π}{3} = - \frac{1}{2}$ . Spotting that $sin x cos x = \frac{1}{2} sin 2 x$ gets there even faster.

Problem 7Numeric answer

Let $f (x) = 2 x^{3} - \frac{4}{x} + 6 x$ . Compute $f^{'} (4)$ as a decimal.

Show the worked solution

Answer: 97.75

I rewrite before I differentiate: $f (x) = 2 x^{3} - 4 x^{- 1} + 6 x^{1/2}$ . The power rule term by term gives $f^{'} (x) = 6 x^{2} + 4 x^{- 2} + 3 x^{- 1/2} = 6 x^{2} + \frac{4}{x ^{2}} + \frac{3}{x}$ . At $x = 4$ : $6 \cdot 16 = 96$ , $\frac{4}{16} = 0.25$ , $\frac{3}{2} = 1.5$ , total $97.75$ . Check the middle term's sign: $- 4 \cdot (- 1) = + 4$ , so the reciprocal term is added, not subtracted.

Problem 8Multiple choice

What is the derivative of $y = 3 x^{2} + 1$ ?

A $\frac{3 x}{3 x ^{2} + 1}$
B $\frac{1}{2 3 x ^{2} + 1}$
C $\frac{6 x}{3 x ^{2} + 1}$
D $3 x 3 x^{2} + 1$

Show the worked solution

Answer: A — $\frac{3 x}{3 x ^{2} + 1}$

The inside is $3 x^{2} + 1$ and the outside is a square root, which I write as a power: $y = (3 x^{2} + 1)^{1/2}$ . Chain rule: outside first, $\frac{1}{2} (3 x^{2} + 1)^{- 1/2}$ , times the inside's derivative $6 x$ . The $6 x$ and the $\frac{1}{2}$ combine to $3 x$ , so $y^{'} = \frac{3 x}{3 x ^{2} + 1}$ . Check at $x = 1$ : the formula gives $\frac{3}{2}$ , and $y (1.01) - y (0.99) \approx 0.03$ over a step of $0.02$ , slope about 1.5.

Problem 9Numeric answer

Let $f (x) = x (2 x + a)^{3}$ , where $a$ is a real constant. Given that $f^{'} (0) = 27$ , find $a$ .

Show the worked solution

Answer: 3

Product rule with $u = x$ and $v = (2 x + a)^{3}$ , where $v^{'}$ needs the chain rule: $v^{'} = 3 (2 x + a)^{2} \cdot 2 = 6 (2 x + a)^{2}$ . So $f^{'} (x) = (2 x + a)^{3} + 6 x (2 x + a)^{2}$ . At $x = 0$ the second term vanishes and the first is $a^{3}$ , so $a^{3} = 27$ and $a = 3$ , the only real cube root. Check: $f^{'} (x) = (2 x + 3)^{2} (8 x + 3)$ after factorising, and at $x = 0$ that is $9 \cdot 3 = 27$ .

Problem 10Numeric answer

Let $f (x) = (x^{2} - 3 x) e^{x}$ . The equation $f^{'} (x) = 0$ has two solutions. Give their sum.

Show the worked solution

Answer: 1

Product rule: $f^{'} (x) = (2 x - 3) e^{x} + (x^{2} - 3 x) e^{x} = e^{x} (x^{2} - x - 3)$ . Since $e^{x} > 0$ for every $x$ , the derivative is zero exactly when $x^{2} - x - 3 = 0$ . The discriminant is $1 + 12 = 13 > 0$ , so there are two real solutions, $x = \frac{1 \pm 13}{2}$ , and their sum is $- \frac{b}{a} = 1$ without computing either root. Check: $\frac{1 + 13}{2} + \frac{1 - 13}{2} = \frac{2}{2} = 1$ .

OMPT differentiation rules: questions students ask

Is the chain rule on the OMPT-A?

Yes. The OMPT-A Differentiation chapter covers the power rule, products, quotients and the chain rule, applied to polynomials, roots, exponentials and logarithms. The typical item is a bracket raised to a power, or e to the power of an expression, with the answer asked as a value at a point. Expect roughly 5 to 7 of the 52 questions from this chapter. Trigonometric derivatives are not on the OMPT-A.

How many differentiation questions are on the OMPT-B?

Differentiation is one of ten chapters on the 64-question OMPT-B, so roughly 5 to 8 questions come from it directly, and the integration chapter leans on it too. The OMPT-B adds sine and cosine to the functions the OMPT-A uses, so a squared sine under the chain rule, or a product of a polynomial and an exponential, are fair game.

Do I need a calculator for differentiation on the OMPT?

No, and none is provided on the maths variants unless a question supplies one. Derivatives are asked at points chosen so that the arithmetic stays small, such as x equals 1, where a logarithm vanishes and a bracket collapses to a single number. If you are multiplying three-digit numbers, you have probably skipped a simplification such as factorising the derivative first.

What is the difference between the product rule and the chain rule on the OMPT?

The product rule handles two things multiplied side by side, such as x squared times e to the x: differentiate each factor in turn and add the two results. The chain rule handles one thing inside another, such as a bracket raised to a power: differentiate the outside, keep the inside, then multiply by the derivative of the inside. Many OMPT items need both, so name the structure before you start.

How do I differentiate a fraction like 3 over x squared for the OMPT?

Rewrite it as a power first: 3 over x squared is 3 times x to the minus 2. The power rule then gives minus 6 times x to the minus 3, which is minus 6 over x cubed. The two slips I mark are keeping the exponent at 2 and losing the minus sign. The same rewrite turns a square root into x to the one half before you differentiate.

Test hubs that cover differentiation rules:OMPT-A OMPT-B OMPT-D OMPT-F All 28 topics

OMPT differentiation rules practice: lesson and 10 problems

OMPT differentiation rules: the lesson

What the OMPT tests in differentiation rules

Notation and answer format the OMPT expects

Worked example 1: the power rule after a rewrite

Worked example 2: the chain rule, narrated

How each OMPT variant tests differentiation rules

OMPT differentiation rules practice problems

OMPT differentiation rules: questions students ask

Related OMPT topics and test hubs