Differentiate xⁿ, for rational values of n, and related constant multiples, sums and differences; differentiate eˢˣ and aˣ, sin kx, cos kx, tan kx and related sums, differences and constant multiples; understand and use the derivative of ln x

This topic covers the differentiation of a wide range of functions, including powers of x (with rational exponents), exponential functions (eˢˣ and aˣ), trigonometric functions (sin kx, cos kx, tan kx), and natural logarithms (ln x). You will learn how to apply the standard rules for constant multiples, sums, and differences, enabling you to differentiate composite expressions efficiently. Mastery of these techniques is essential for solving problems in calculus, such as finding gradients, rates of change, and optimisation, and they form the foundation for more advanced topics like integration and differential equations.

In the Edexcel A-Level Mathematics specification, this content appears in both Pure Mathematics 1 and Pure Mathematics 2. It builds on the basic rules of differentiation (power rule, constant multiple rule, sum/difference rule) and extends them to new functions. Understanding the derivative of ln x is particularly important as it links to integration of 1/x and to solving differential equations. The ability to differentiate eˢˣ and aˣ is crucial for modelling exponential growth and decay, while trigonometric differentiation is essential for analysing periodic behaviour in contexts such as physics and engineering.

By the end of this topic, you should be able to differentiate any combination of these functions confidently, applying the chain rule where necessary (though chain rule is covered separately, it is often used in conjunction with these rules). This knowledge is directly assessed in exam questions, often as part of multi-step problems involving tangents, normals, stationary points, or connected rates of change.