Tangents and derivatives

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In the Euclidean plane with a coordinate system, let (C) be a curve with equation f (x) = 0, and let M₀ and M be two points on the curve, with respective x-coordinates x₀ and x. The slope of the secant line (M₀M) is given by the difference quotient between M₀ and M: it is equal to $\frac{f(x)-f(x_0)}{x-x_0}\backslash frac\{f(x)-f(x\_0)\}\{x-x\_0\}$.

As M approaches M₀, the secant line (M₀M) generally tends toward the tangent line to (C) at M₀ (shown in blue on the graph). Even without defining this limit precisely, we can see from the graph that the line (M₀M') is "closer" to the tangent at M₀ than the line (M₀M), because M' is "closer" to M₀ than M is. Special cases can arise at points where there is no tangent, or for particularly "pathological" curves.

The slope of the tangent is generally the limit of the slope of the line (M₀M)—that is, the limit of the difference quotient between M₀ and M as M approaches M₀, or equivalently as x approaches x₀. This limit is the derivative of f at x₀: