Derivative Formulas for Grades 11 and 12: A Quick-Reference Table and How to Understand Them Many students know their derivative formulas by heart but still get questions wrong in the exam. The reason usually isn’t a lack of formulas. It’s that they have learnt them by rote, and then freeze when a question appears in an unfamiliar form. This article brings together all the derivative formulas for Grades 11 and 12 in a quick-reference table. More importantly, you will understand why each formula works the way it does. Once you understand the underlying ideas, you will find it easier to handle familiar question types and the variations that often come up. A complete set of derivative formulas for Grades 11 and 12, with an approach to understanding them in depth What is a derivative? Understanding its meaning before learning the formulas Definition of a derivative The derivative of a function measures the rate at which the function changes at a given point. Imagine a moving car. The distance travelled changes over time, and the speed is the derivative of the distance. Mathematically, the derivative is the limit of the difference quotient. As the change in the variable approaches 0, that ratio gives us the derivative. In many quick calculations you don’t need to use the limit definition directly, but it is still worth understanding what it means. Exploring the definition of a derivative in basic mathematics Geometric and physical meaning On a graph, the derivative at a point is the gradient of the tangent at that point. If the tangent slopes upwards, the derivative is positive. If the tangent is horizontal, the derivative is 0. This is why derivatives appear in problems about maxima and minima. If a function is differentiable at a turning point inside the interval being considered, its derivative there is 0. In physics, the derivative of displacement is velocity, and the derivative of velocity is acceleration. One tool, many different applications. Complete table of derivative formulas This is the section you will come back to most often. The formulas are grouped to make them easy to find. Derivatives of basic functions Function Derivative c (constant) 0 x 1 xⁿ n · xⁿ⁻¹ √x 1 / (2√x), for x > 0 1/x -1 / x², for x ≠ 0 eˣ eˣ aˣ aˣ · ln a ln x 1/x, for x > 0 logₐ x 1 / (x · ln a), for x > 0, a > 0, a ≠ 1 Rules of differentiation The four rules below cover most exercises that combine functions. Sum and difference: (u ± v)′ = u′ ± v′ Constant multiple: (k·u)′ = k·u′ Product: (u·v)′ = u′·v + u·v′ Quotient: (u/v)′ = (u′·v − u·v′) / v² A complete, systematic summary of differentiation rules and formulas Derivatives of trigonometric functions Function Derivative sin x cos x cos x -sin x tan x 1 / cos²x, for cos x ≠ 0 cot x -1 / sin²x, for sin x ≠ 0 Note the minus signs for cos x and cot x. This is where a lot of students go wrong. Easy-to-remember derivative formulas for trigonometric functions Derivatives of composite functions (the chain rule) A composite function is a function inside another function, for example sin(2x + 1). The chain rule tells us to multiply the derivative of the outer function by the derivative of the inner function. If y = f(u) and u = u(x), then: y′ₓ = f′(u) · u′(x) Common cases when differentiating composite functions: (uⁿ)′ = n·uⁿ⁻¹·u′ (√u)′ = u′ / (2√u) (sin u)′ = cos u · u′ (cos u)′ = −sin u · u′ (eᵘ)′ = eᵘ · u′ (ln u)′ = u′ / u (aᵘ)′ = aᵘ · ln a · u′ (logₐ u)′ = u′ / (u·ln a) Higher-order derivatives The second derivative is the derivative of the first derivative, written f″(x). In the same way, we have third and fourth derivatives. The second derivative is used to determine the concavity of a graph. It can also help check for maxima and minima in many cases, but it needs to be combined with the first derivative and a sign analysis. How to differentiate: step-by-step examples The formulas in this article are organised based on the derivatives content of the Grade 11 and 12 Mathematics curriculum and checked against foundational calculus resources such as OpenStax Calculus. When studying, students should also pay attention to the domain of each function, to avoid applying formulas mechanically. A chain rule example – where students go wrong most often Find the derivative of: y = (3x² + 1)⁵ Step 1. Identify the inner and outer functions. The inner function is u = 3x² + 1. The outer function is u⁵. Step 2. Differentiate the inner function. u′ = 6x. Step 3. Apply the formula (uⁿ)′ = n·uⁿ⁻¹·u′. This gives y′ = 5·(3x² + 1)⁴·6x. Step 4. Simplify. y′ = 30x·(3x² + 1)⁴. The easiest place to go wrong is step 3. Many students write y′ = 5·(3x² + 1)⁴ and stop there. They forget to multiply by u′ = 6x, and the answer is completely wrong. Let’s try another short example with a trigonometric function. For y = sin(2x + 1), we get y′ = cos(2x + 1)·2 = 2cos(2x + 1). A detailed guide to differentiation through worked examples 5 common mistakes when differentiating and how to avoid them The mistakes below come up again and again in tests. Know them in advance so you can avoid them. Forgetting to multiply by the derivative of the inner function. This is the number one mistake with composite functions. Always ask yourself: have I multiplied by u′? Getting the signs wrong for cos and cot. The derivative of cos x is −sin x, not sin x. Applying the quotient rule incorrectly. The numerator is u′·v − u·v′; the order must not be reversed. Dropping the constant multiple. For y = 3x⁴, the derivative is 12x³, not 4x³. Getting the power wrong. For xⁿ, the new power is n − 1, and it is easy to subtract incorrectly. A quick tip: once you have finished, test your answer with a specific value of x. This helps you spot mistakes quickly. Common mistakes that cost marks when differentiating, and how to avoid them Derivatives in international curricula and their applications beyond Grade 12 In the IB Diploma Programme, Mathematics: analysis and approaches helps students develop their skills in calculation, mathematical reasoning and proof. Mathematics: applications and interpretation, by contrast, focuses on using derivatives to solve real-world problems, analyse data and build mathematical models. Compared with the Vietnamese textbook curriculum, the approach in international curricula is somewhat different. Students in Vietnam typically start from formulas and practise many question types. In many international curricula, such as the IB DP, students are expected to develop logical and critical thinking, generalisation and the use of technology in mathematics. Derivatives in international curricula and future learning pathways Beyond Grade 12, derivatives continue to be applied in many academic and professional fields. In economics, derivatives help optimise profit and cost. In engineering, they are used in design, control and systems analysis. In data science, derivatives underpin many machine learning algorithms. So if your child is planning to study abroad or follow an international curriculum, a solid foundation in calculus will be an important advantage. Parents can find out more about the IB programmes at Westlink to better understand how Mathematics is taught with a focus on thinking, application and an international outlook. Tips for remembering and practising derivatives effectively Learning derivatives isn’t about memorising a long list. Here are a few ways to remember them for longer and apply them well. First, understand before you memorise. Once you understand that a derivative is a gradient, you won’t forget what it means. The formulas then become just tools. Second, group formulas by family. Trigonometric functions go together. Exponential and logarithmic functions go together. Learning in groups is easier than learning formulas in isolation. Third, practise a little every day. Ten minutes a day is better than three hours crammed in at the weekend. Differentiation is a skill, and it needs repetition to become second nature. Finally, redo the questions you got wrong. Every mistake is a lesson. Once you understand why you got something wrong, you won’t repeat it. Tips for remembering derivative formulas by function family Learning Mathematics for understanding at Westlink Westlink is an IB World School authorised for the Primary Years Programme (PYP) and the Middle Years Programme (MYP), and is a candidate school for the Diploma Programme (DP). Students are encouraged to ask questions and explore the reasoning behind a problem rather than simply memorising formulas. Mathematical concepts are explained in terms of why they work, how to apply them and how they connect to real life. Thanks to Westlink’s international IB approach, this inquiry- and research-based way of learning is applied consistently across the different stages of schooling. Through this approach, students have the opportunity to develop logical thinking, analytical skills and problem-solving skills. Rather than simply looking for the right answer, they are encouraged to explain their reasoning and justify their answers. This also gives students a foundation for approaching problems with more confidence when there are several possible methods or the situation is new. Mastering derivative formulas doesn’t just help you do well in tests; it also lays an important foundation for studying mathematics at university and in international programmes. Remember the formulas, understand the ideas behind them and know how to apply them to each type of question. If you would like your child to develop well-rounded mathematical thinking and grow in confidence in an international learning environment, contact Westlink today for advice on the most suitable learning pathway for your child.
