Tangent Line Calculator
The tangent line to a polynomial at a chosen x — its slope, equation and the normal line — from the derivative evaluated at that point.
The slope of the tangent at x₀ is the derivative there, f′(x₀).
How the tangent line calculator works
The slope of the tangent at x₀ is the derivative there, f′(x₀). With the point (x₀, f(x₀)) the line is y = f(x₀) + f′(x₀)(x − x₀). The normal is perpendicular, with slope −1 ÷ f′(x₀). Enter coefficients from the highest power down, with zeros for missing terms.
Formula: y = f(x₀) + f′(x₀)(x − x₀); normal slope = −1 ÷ f′(x₀)
Worked examples
| Inputs | Slope of the tangent | Note |
|---|---|---|
| x² − 3 at x = 2 | 4 | slope 4; y = 4x − 7 |
| x³ at x = 1 | 3 | slope 3; y = 3x − 2 |
| A flat point: x² at x = 0 | 0 | slope 0; horizontal tangent |
FAQFrequently asked questions
What does the tangent tell me?
The direction the curve is heading at that instant — the best straight-line approximation nearby. Its slope is the instantaneous rate of change, which is what a derivative is.
What is the normal for?
The line at right angles to the curve at the point — used for reflections, for the direction a surface pushes back, and in optimisation to find the closest point on a curve.
Why is the normal vertical when the slope is zero?
At a flat point the tangent is horizontal, and perpendicular to horizontal is vertical: x = x₀. Its slope would be −1 ÷ 0, which is why it is written as a vertical line.
Can I use this for non-polynomials?
Not directly — enter a polynomial fitted to your function near the point, or use the derivative you know: the line is y = f(x₀) + f′(x₀)(x − x₀) for any differentiable function.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.