Pure · Differentiation
Chapter 1 · 3
The idea
Tangents and normals
How to find the equation of the tangent and the normal to a curve at a point: get the gradient from the derivative, the point from the curve, then write a straight line — and why the normal's gradient is the negative reciprocal.
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Pure · Differentiation
Tangents and normals
How to find the equation of the tangent and the normal to a curve at a point: get the gradient from the derivative, the point from the curve, then write a straight line — and why the normal's gradient is the negative reciprocal.
Why it works
A point and a gradient
A tangent is just a straight line that grazes the curve at one point, matching its direction there. So to pin it down you only need the two things any straight line needs: a point it passes through, and a gradient.- The point lies on the curve: at it is — substitute into the original equation to get the height.
- The gradient is the steepness of the curve at that point — which is exactly what the derivative gives: . Differentiate, then substitute.
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