↗️ Tangent Line Calculator
The tangent line is the straight line that just touches a curve at one point and travels in exactly the curve’s direction there. Its slope is the derivative at that point — which is why the tangent line is where derivatives stop being abstract and become geometry.
Type any function and the x-coordinate of the point of tangency. This calculator differentiates the function symbolically (the same engine as our Derivative Calculator), evaluates f and f′ at your point, and writes the tangent line and the normal line, with the curve and its tangent drawn.
How to use this calculator
The recipe never changes: slope = f′(a), point = (a, f(a)), then point-slope form.
- Type your function f(x) — polynomials, trig, exponentials, logs and roots all work.
- Enter the x-coordinate a of the point of tangency.
- Press Calculate.
- Read the tangent line equation, and check it against the graph: the red line should just kiss the curve at the marked point.
Tangent Line Worked Example
Find the tangent to f(x) = x² at x = 1. The derivative is f′(x) = 2x, so the slope at the point is f′(1) = 2, and the point itself is (1, 1). Point-slope form: y − 1 = 2(x − 1), which tidies to y = 2x − 1. The normal line, perpendicular at the same point, is y = −1/2x + 3/2. These are the defaults above — the graph shows the parabola and its tangent touching at (1, 1).
Frequently asked questions
How is a tangent line calculated?
Three steps: evaluate the function at the point to get (a, f(a)); differentiate and evaluate to get the slope m = f′(a); then write y − f(a) = m(x − a) and rearrange. This calculator performs all three with an exact symbolic derivative, not a numerical approximation.
What is the normal line?
The normal is the line perpendicular to the tangent at the same point. Its slope is the negative reciprocal of the tangent’s slope, −1/m. Normals matter in optics (reflection), physics (forces on surfaces) and geometry (the normal through a circle’s centre hits the point of tangency).
Can a tangent line cross the curve?
Yes — touching at one point does not mean staying on one side. At an inflection point (for example f(x) = x³ at x = 0) the tangent crosses the curve. It still has the curve’s exact slope there; "tangent" is about direction, not about never crossing.
What if the derivative does not exist at the point?
Sharp corners (like |x| at 0), cusps and vertical tangents have no ordinary tangent line, and the calculator will tell you the derivative is not finite there instead of inventing an answer.