When the curve is given by "y" = "f"("x") then the slope of the tangent is formula_4 so by the point–slope formula the equation of the tangent line at ("X", "Y") is

where ("x", "y") are the coordinates of any point on the tangent line, and where the derivative is evaluated at formula_6.

When the curve is given by "y" = "f"("x"), the tangent line's equation can also be found by using polynomial division to divide formula_7 by formula_8; if the remainder is denoted by formula_9, then the equation of the tangent line is given by

When the equation of the curve is given in the form "f"("x", "y") = 0 then the value of the slope can be found by implicit differentiation, giving

The equation of the tangent line at a point ("X","Y") such that "f"("X","Y") = 0 is then

This equation remains true if formula_13 but formula_14 (in this case the slope of the tangent is infinite). If formula_15 the tangent line is not defined and the point ("X","Y") is said singular.For algebraic curves, computations may be simplified somewhat by converting to homogeneous coordinates. Specifically, let the homogeneous equation of the curve be "g"("x", "y", "z") = 0 where "g" is a homogeneous function of degree "n". Then, if ("X", "Y", "Z") lies on the curve, Euler's theorem impliesformula_16It follows that the homogeneous equation of the tangent line is

The equation of the tangent line in Cartesian coordinates can be found by setting "z"=1 in this equation.

To apply this to algebraic curves, write "f"("x", "y") as

where each "u" is the sum of all terms of degree "r". The homogeneous equation of the curve is then

Applying the equation above and setting "z"=1 produces

as the equation of the tangent line. The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.

If the curve is given parametrically by

then the slope of the tangent is

giving the equation for the tangent line at formula_23 as

If formula_25 the tangent line is not defined. However, it may occur that the tangent line exists and may be computed from an implicit equation of the curve.

Multiple tangents at a point
Tangent line to a curve
Intuitive description
Equations