Find Tangent Line Equations
Calculate the equation of a tangent line to any function at a specified point. Our calculator finds the slope using derivatives and provides the line equation in slope-intercept form, essential for calculus and curve analysis.
Enter a function and a point to calculate the equation of the tangent line at that point.
A tangent line to a curve at a given point is a straight line that just touches the curve at that point. It has the same slope as the curve at that point and represents the instantaneous rate of change of the function.
To find the equation of the tangent line to the function f(x) at x = a:
Tangent lines are used in various fields such as physics for motion analysis, in economics for marginal analysis, and in engineering for approximations.
A tangent line touches a curve at exactly one point and has the same slope as the curve at that point. The slope equals the derivative f'(x) evaluated at the point. Tangent line equation: y - y₁ = m(x - x₁), where m = f'(x₁) and (x₁, y₁) is the point of tangency.
Tangent lines are used for: linear approximation of functions near a point, understanding instantaneous rate of change, Newton's method for finding roots, optimization problems, physics motion analysis, and estimating function values. They represent the best linear approximation to a curve locally.
Normal line: perpendicular to tangent (slope = -1/m). Secant line: intersects curve at two points (average rate of change). As secant points converge, it becomes the tangent. The derivative is the limit of secant slopes as the distance between points approaches zero.
