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Tangent Line Calculator

Find the tangent line to any curve at any point — see it drawn live

The tangent line to a curve at a point is the straight line that just touches the curve at that point, matching its slope exactly. The slope of the tangent equals the derivative f'(a) at that point.

Given a function f(x) and a point x = a, the tangent line equation is:

y = f(a) + f'(a)(x - a)

This tangent line calculator with graph draws the tangent visually on an interactive graph — see the line, the point of tangency, and the slope in real time. Type any function and a point, and the AI computes the exact tangent line using numerical differentiation. Zoom, pan, and explore the relationship between the curve and its tangent.

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FAQ

What is a tangent line?
A tangent line is a straight line that touches a curve at exactly one point and has the same slope as the curve at that point. The slope of the tangent line equals the derivative f'(a) at the point of tangency.
How do I find the equation of a tangent line?
Use the point-slope form: y - f(a) = f'(a)(x - a). You need two things: the y-value f(a) and the slope f'(a) at your point. This calculator computes both numerically for any function.
What is the difference between a tangent line and a normal line?
The tangent line has slope f'(a). The normal line is perpendicular to the tangent, so its slope is -1/f'(a). Together they form a right angle at the point of tangency.
Can I find the tangent to sin(x), e^x, or other functions?
Yes. This calculator uses numerical differentiation, so it works with any function — polynomial, trigonometric, exponential, logarithmic, or any combination.
What if the tangent line is vertical?
A vertical tangent occurs when f'(a) is undefined or infinite (e.g., at x = 0 for y = x^{1/3}). The calculator will detect this and tell you the tangent is vertical.
Why use a tangent line calculator with a graph?
A graph makes the tangent line visual and intuitive. You can see how the tangent touches the curve at exactly one point, how its slope matches the curve's direction, and how it changes as you pick different points. Interactive zoom and pan let you explore the relationship in detail — much better than just a number.

Related Topics

Derivatives (lesson)Root FinderSlope of a Line