Apply differentiation to find gradients, tangents and normals; maxima and minima and stationary points; points of inflection; identify where functions are increasing or decreasing

Differentiation is a fundamental tool in calculus that allows us to analyse the rate of change of functions. In A-Level Mathematics (Edexcel), you will learn to apply differentiation to find gradients of curves at any point, which is essential for determining the equations of tangents and normals. This skill is crucial for understanding the behaviour of functions, including where they are increasing or decreasing, and for locating stationary points (maxima, minima, and points of inflection). These concepts are widely used in optimisation problems across physics, economics, and engineering, making them a core part of the curriculum.

The topic builds on the basic rules of differentiation (power, product, quotient, and chain rules) and extends to applications such as curve sketching and solving real-world problems. You will learn to identify stationary points by setting the first derivative to zero, and then classify them using the second derivative or a sign diagram. Points of inflection occur where the second derivative changes sign, indicating a change in concavity. Understanding these concepts allows you to fully describe the shape of a function and its turning points, which is essential for higher-level mathematics and many STEM fields.

Mastering this topic is vital for exam success, as questions often require you to find gradients, write equations of tangents and normals, and determine the nature of stationary points. You will also need to interpret the first and second derivatives to describe intervals where a function is increasing or decreasing. These skills are tested in both pure mathematics and applied contexts, such as kinematics and optimisation. By the end of this topic, you should be able to confidently analyse any differentiable function and communicate its key features.