Tangent Line Calculator

What Is a Tangent Line?

A tangent line to a curve at a point is the straight line that touches the curve there and points in the same direction as the curve at that instant. If you zoom in far enough on a smooth curve, it becomes almost indistinguishable from its tangent line.

The tangent line to y = x² at x = 3 passes through (3, 9) with slope 6, so its equation is y = 6x − 9.

How to Find a Tangent Line

1. Evaluate f(a) to find the point of tangency (a, f(a)).

2. Differentiate to find f′(x).

3. Evaluate f′(a); this is the slope m.

4. Substitute into y − y₁ = m(x − x₁), then simplify to y = mx + b if you need slope-intercept form.

For f(x) = x³ − 2x at x = 1: f(1) = −1, f′(x) = 3x² − 2, f′(1) = 1, so y + 1 = 1(x − 1), which simplifies to y = x − 2.

Tangent Line Formula

y − y₁ = m(x − x₁), with x₁ = a, y₁ = f(a) and m = f′(a). Together: y − f(a) = f′(a)(x − a).

Solved for y, this is y = f(a) + f′(a)(x − a), which is also the formula for the linear approximation L(x).

How the Derivative Gives the Slope

The slope of a secant line through (a, f(a)) and (a + h, f(a + h)) is [f(a + h) − f(a)]/h. As h → 0, the second point slides along the curve toward the first, and the secant turns into the tangent. The limit of those slopes is the derivative: f′(a) = lim(h→0) [f(a + h) − f(a)]/h.

So the derivative is exactly the slope of the tangent line: for f(x) = x², f′(x) = 2x, and the tangent at x = 3 has slope f′(3) = 6.

How to Find the Point of Tangency

The point of tangency is on the curve, so its y-coordinate is the function value: y₁ = f(a). A common mistake is to use f′(a) as the y-coordinate. f′(a) is the slope, not a height.

For f(x) = √x at x = 4, the point is (4, √4) = (4, 2), while the slope is f′(4) = 1/(2√4) = 1/4.

Point-Slope Form

A line through (x₁, y₁) with slope m can be written y − y₁ = m(x − x₁). It is the natural form for tangent lines because the calculation already gives a point and a slope.

To convert to slope-intercept form, expand and solve for y. For y − 9 = 6(x − 3): y − 9 = 6x − 18, so y = 6x − 9. Both equations describe the same line.

Tangent Line Examples

f(x) = sin(x) at x = 0: the point is (0, 0), f′(x) = cos(x) and f′(0) = 1, so the tangent is y = x.

f(x) = eˣ at x = 1: the point is (1, e), f′(x) = eˣ and the slope is e, so y − e = e(x − 1), which simplifies to y = ex.

f(x) = √x at x = 4: the point is (4, 2), f′(x) = 1/(2√x) and the slope is 1/4, so y − 2 = (1/4)(x − 4), or y = (1/4)x + 1.

f(x) = 1/x at x = 2: the point is (2, 1/2), f′(x) = −1/x² and the slope is −1/4, so y = (−1/4)x + 1.

Tangent Line vs Secant Line

A secant line passes through two points on a curve, and its slope is the average rate of change between them. A tangent line uses a single point, and its slope is the instantaneous rate of change there, given by the derivative.

As the two points of a secant move together, the secant approaches the tangent line. That limiting process is how the derivative is defined.

Horizontal and Vertical Tangents

A horizontal tangent occurs where f′(a) = 0. The line is y = f(a). For f(x) = x² − 4x + 1 at x = 2, f′(2) = 0 and the tangent is y = −3. Horizontal tangents mark possible maximums and minimums.

A vertical tangent occurs where the slope grows without bound, as for f(x) = ∛x at x = 0. Its equation is x = a; it has no slope and cannot be written as y = mx + b.

At a corner, such as |x| at x = 0, the slopes from the left and right differ, and there is no tangent line at all.

Tangent Lines and Linear Approximation

Near x = a, a differentiable function is close to its tangent line. Written as a function, the tangent line is the linearization L(x) = f(a) + f′(a)(x − a), and f(x) ≈ L(x) for x near a.

For example, with f(x) = √x at a = 4, L(x) = 2 + (1/4)(x − 4), so √4.1 ≈ 2 + 0.025 = 2.025. The true value is about 2.02485. The approximation is best close to a and gets worse farther away.

Applications of Tangent Lines

Tangent lines give instantaneous rates of change: the tangent to a position–time graph gives velocity, and the tangent to a cost curve gives marginal cost. They are the basis of linear approximation and of Newton's method for solving equations, which follows tangent lines to the x-axis. They also help locate maximums and minimums, where the tangent is horizontal.