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A tangent touches a curve with the same gradient at a point, and the normal is perpendicular to it. This topic uses differentiation to find the gradient at a point and the equations of the tangent and normal. It appears across A-Level and iAL Pure Maths with the major exam boards. Learn it free on eClassroom through guided worked examples and instantly-marked exam-style questions.
How you do it
Differentiate and substitute the x value to get the gradient at that point. The tangent uses that gradient; the normal uses its negative reciprocal. Then put the point and the gradient into y minus y1 equals m times x minus x1.
Before you start
It draws heavily on basic differentiation and straight-line equations. Keep it sharp for curve sketching and optimisation. It belongs to the Pure 1 specification.
What you learn
A tangent just touches a curve, matching its gradient at that point; a normal cuts it at right angles. Differentiation gives the gradient, and then it’s straight-line geometry from there. The tangent at a point on a curve has the same gradient as the curve there. Don’t mix them up — the point comes from the original curve, not the derivative. The normal is perpendicular to the tangent at the point of contact. The course works through the gradient of a tangent, equation of a tangent, equation of a normal, where tangents and normals meet the axes and finding the point from a gradient. Every section has instant-feedback practice questions matched to Edexcel International A-Level and 9MA0 Pure, and the course finishes with full exam-style questions you mark yourself.
Related Topics
We teach Tangents & Normals as one of the Differentiation topics in our A-level course library, at the AS year. The course is built from 5 concept sections, 5 worked examples and 5 check points before the exam-style set. It sits alongside Basic Differentiation, Chain Rule, Connected Rates of Change and Differentiation from First Principles in the same strand. You can drill this strand further on the Differentiation revision engine, with your accuracy saved to your skill tree.
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