Tangents and normals
Objective. Construct equations using derivative gradients.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A tangent to a curve at a point touches it there and has the same gradient as the curve, found by differentiating and substituting the x-coordinate, while the normal is perpendicular to the tangent at that same point. If the tangent gradient is m, the normal gradient is −1/m, since perpendicular lines have gradients that multiply to give −1. Using the point and gradient in y − y1 = m(x − x1) gives the equation of either line. For example, for y = x² at x = 2, the gradient is 2x = 4, so the tangent is y − 4 = 4(x − 2), while the normal uses gradient −1/4.
Worked example
Find the equation of the normal to the curve y = x² − 3x at the point where x = 1.
- 1Differentiate: dy/dx = 2x − 3, so at x = 1 the gradient is −1.
- 2Find y at x = 1: y = 1 − 3 = −2, giving point (1, −2).
- 3Normal gradient is the negative reciprocal: −1/(−1) = 1.
- 4Substitute into y − (−2) = 1(x − 1) to get y = x − 3.
Answer: y = x − 3
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Practice
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Start practisingCommon mistake. Students sometimes use the curve's gradient formula directly as the normal's gradient instead of taking its negative reciprocal. Always compute −1/m for the normal, never m itself or just −m.
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