Enter a polynomial and a point on it, and instantly get the equations of the tangent and normal lines at that point.
Tangent & Normal Line Calculator
How It’s Calculated
The tangent line at x = a has slope f'(a) — the derivative evaluated at that point — and passes through (a, f(a)), giving y − f(a) = f'(a)(x − a). The normal line is perpendicular to the tangent at the same point, so its slope is the negative reciprocal, −1/f'(a) — unless the tangent is horizontal (f'(a) = 0), in which case the normal is the vertical line x = a.
Worked Example
For f(x) = x² at a = 2: f(2) = 4, f'(x) = 2x so f'(2) = 4. Tangent: y − 4 = 4(x − 2), i.e. y = 4x − 4. Normal slope = −1/4, so Normal: y − 4 = −0.25(x − 2).
Frequently Asked Questions
Q: Why is the normal line’s slope the negative reciprocal?
Two lines are perpendicular exactly when the product of their slopes is −1 — that’s the defining geometric relationship, and it’s where −1/f'(a) comes from.
Q: What happens if f'(a) = 0?
The tangent is horizontal (y = f(a)), and the normal — perpendicular to a horizontal line — is vertical (x = a). A negative reciprocal of 0 is undefined, which is exactly why the vertical case needs separate handling.
Tangent and normal lines are a core Application of Derivatives topic in 2nd PUC. See how Daniel Classes teaches 1st and 2nd PUC Mathematics →