Differentiation — OCR A-Level Mathematics

Differentiation is a fundamental concept in calculus that deals with rates of change and the slopes of curves. In OCR A-Level Mathematics, you will learn how to differentiate a wide range of functions, including polynomials, exponentials, logarithms, and trigonometric functions. The derivative of a function gives the gradient of the tangent at any point, which is essential for solving optimisation problems, finding stationary points, and analysing motion in mechanics.

This topic is crucial because it underpins many areas of mathematics and physics. In pure mathematics, differentiation allows you to sketch curves accurately by identifying turning points and points of inflection. In applied mathematics, it is used to model velocity and acceleration from displacement functions. Mastery of differentiation is also a prerequisite for integration, which you will study later in the course.

Differentiation builds on your knowledge of algebra and coordinate geometry. You will need to be comfortable with algebraic manipulation, including expanding brackets and simplifying expressions, as well as understanding the concept of limits. The OCR specification covers both first and second derivatives, and you will be expected to apply differentiation to real-world contexts, such as economics and engineering.

• →The derivative as the gradient of a tangent and as a rate of change.
• →Differentiating polynomials using the power rule: d/dx(x^n) = n x^(n-1).
• →Differentiating exponentials (e^x), natural logarithms (ln x), and trigonometric functions (sin x, cos x, tan x).
• →The chain rule, product rule, and quotient rule for differentiating composite, product, and quotient functions.
• →Using second derivatives to determine the nature of stationary points (maxima, minima, points of inflection).

• Correct application of the chain, product, and quotient rules.
• Correct differentiation of standard functions including polynomials, exponentials, logarithms, and trigonometric functions.
• Accurate determination of stationary points and their nature using the second derivative.
• Correct formation of equations for tangents and normals.
• Clear and logical presentation of differentiation from first principles.
• Correct use of notation such as dy/dx, f'(x), and f''(x).
• Accurate identification of increasing and decreasing functions using the sign of the derivative.

• 💡Always write down the derivative expression before evaluating it at a specific point.
• 💡Use the calculator to check derivatives where appropriate, but ensure all analytical steps are shown to gain full marks.
• 💡When asked to 'show that', ensure every intermediate step of the differentiation is clearly visible.
• 💡Remember that the gradient of a normal is the negative reciprocal of the gradient of the tangent.
• 💡Check the units and context when solving problems involving rates of change.
• 💡Always simplify your derivative before substituting values. For example, factorising can help avoid algebraic errors when finding stationary points.
• 💡When using the product or quotient rule, clearly label u and v to avoid confusion. Show each step of differentiation to pick up method marks even if your final answer is wrong.
• 💡For optimisation problems, check the domain of the function and consider whether endpoints or stationary points give the maximum or minimum. Don't forget to justify the nature of stationary points using the second derivative or a sign table.

• Confusing the product rule with the quotient rule or misapplying them.
• Failing to use the chain rule when differentiating composite functions.
• Errors in sign when differentiating trigonometric functions (e.g., differentiating cos x to sin x instead of -sin x).
• Incorrectly identifying the nature of stationary points or points of inflection.
• Forgetting the constant of integration when working backwards from a derivative.
• Misinterpreting the difference between plotting and sketching a curve.
• Confusing the derivative of e^x with x e^(x-1). Remember: d/dx(e^x) = e^x, not x e^(x-1).
• Forgetting to apply the chain rule when differentiating composite functions like sin(2x). The derivative is 2 cos(2x), not cos(2x).
• Thinking that a stationary point where the second derivative is zero is always a point of inflection. It could be a maximum or minimum if the second derivative changes sign, so always check the sign on either side.