tangentline

A tangent line to a curve at a given point is the straight line that best approximates the curve near that point. It touches the curve at that point and shares the curve’s local direction there, reflecting the curve’s first-order, or linear, behavior.

If a curve is given as a function y = f(x) and f is differentiable at x = a, the tangent line to the graph at the point (a, f(a)) has equation y = f(a) + f'(a)(x − a). The slope is the derivative at that point and represents the instantaneous rate of change of the function.

For curves given parametrically by r(t) = (x(t), y(t)), the tangent line at t0 is the line through r(t0) in the direction of r'(t0); a common parametrization of the tangent line is L(s) = r(t0) + s r'(t0).

For curves defined implicitly by F(x, y) = 0 with defined partial derivatives, the slope is dy/dx = −F_x/F_y, provided F_y ≠ 0 at the point. A tangent line at a point (x0, y0) with F(x0, y0) = 0 can be written in various equivalent forms, such as y − y0 = m (x − x0) with m = −F_x(x0, y0)/F_y(x0, y0).

Examples include the tangent to y = x^2 at x = 1, which is y = 2x − 1, and the tangent to the circle x^2 + y^2 = 1 at (1, 0), which is the vertical line x = 1. Tangent lines are fundamental in linear approximation and in the local analysis of curves in mathematics, physics, and engineering.