Tangent line

A tangent line is the line through a point on a curve whose direction matches the curve’s direction there. For a differentiable graph, its slope equals the derivative at that point.

Derivative: A derivative gives the instantaneous rate of change of a function with respect to its input. At a differentiable point, the derivative gives the tangent line’s slope.
Slope: A measure of a line’s steepness, given by vertical change divided by horizontal change. A tangent is determined by its slope together with a point on the curve.
Linear approximation: An estimate of a function near a point using its value and derivative at that point. The tangent line supplies the simplest derivative-based estimate of nearby curve values.
Corner (mathematics): A point on a curve where the incoming and outgoing directions differ sharply. A corner can prevent a curve from having one tangent direction.
Difference quotient: A quotient measuring a function’s change over a finite change in its input. Its limit as the interval shrinks produces the slope used for a tangent.
Point-slope form: The equation form y − y₁ = m(x − x₁) for a line of slope m through (x₁, y₁). It directly constructs the tangent from the curve point and derivative.
Newton's method: An iterative method for approximating a root by repeatedly using function values and derivatives. Each update uses the tangent’s x-intercept as the next root estimate.
Cusp (singularity): A sharp singular point where branches of a curve meet with a pointed shape. A cusp may lack a finite tangent slope despite having a limiting tangent direction.
Limit (mathematics): A value that a function or sequence approaches as its input approaches a specified value. A tangent slope is defined through a limiting process.
Secant line: A line intersecting a curve at two or more points. Tangents can be obtained as limiting positions of secants as their points converge.

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