Analytical approach

At each point, the moving line is always tangent to the curve. Its slope is the derivative; green marks positive derivative, red marks negative derivative and black marks zero derivative. The point (x,y) = (0,1) where the tangent intersects the curve, is not a max, or a min, but is a point of inflection. (Note: the figure contains the incorrect labeling of 0,0 which should be 0,1)

The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century. In the second book of his Geometry, René Descartes^[10] said of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know".^[11]

Equations

When the curve is given by y = f(x) then the slope of the tangent is ${\displaystyle dy/dx,}$ so by the point–slope formula the equation of the tangent line at (X, Y) is

${\displaystyle y-Y={\frac {dy}{dx}}(X)\cdot (x-X)}$

where (x, y) are the coordinates of any point on the tangent line, and where the derivative is evaluated at ${\displaystyle x=X}$.^[12]

When the curve is given by y = f(x), the tangent line's equation can also be found^[13] by using polynomial division to divide ${\displaystyle f\,(x)}$ by ${\displaystyle (x-X)^{2}}$; if the remainder is denoted by ${\displaystyle g(x)}$, then the equation of the tangent line is given by

${\displaystyle y=g(x).}$

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

${\displaystyle {\frac {dy}{dx}}=-{\frac {\partial f}{\partial x}}{\bigg /}{\frac {\partial f}{\partial y}}.}$

The equation of the tangent line at a point (X,Y) such that f(X,Y) = 0 is then^[12]

${\displaystyle {\frac {\partial f}{\partial x}}(X,Y)\cdot (x-X)+{\frac {\partial f}{\partial y}}(X,Y)\cdot (y-Y)=0.}$

This equation remains true if

${\displaystyle {\frac {\partial f}{\partial y}}(X,Y)=0,\quad {\frac {\partial f}{\partial x}}(X,Y)\neq 0,}$

in which case the slope of the tangent is infinite. If, however,

${\displaystyle {\frac {\partial f}{\partial y}}(X,Y)={\frac {\partial f}{\partial x}}(X,Y)=0,}$

the tangent line is not defined and the point (X,Y) is said to be 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 implies

$${\displaystyle {\frac {\partial g}{\partial x}}\cdot X+{\frac {\partial g}{\partial y}}\cdot Y+{\frac {\partial g}{\partial z}}\cdot Z=ng(X,Y,Z)=0.}$$

It follows that the homogeneous equation of the tangent line is

${\displaystyle {\frac {\partial g}{\partial x}}(X,Y,Z)\cdot x+{\frac {\partial g}{\partial y}}(X,Y,Z)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,Z)\cdot z=0.}$

The equation of the tangent line in Cartesian coordinates can be found by setting z=1 in this equation.^[14]

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

${\displaystyle f=u_{n}+u_{n-1}+\dots +u_{1}+u_{0}\,}$

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

${\displaystyle g=u_{n}+u_{n-1}z+\dots +u_{1}z^{n-1}+u_{0}z^{n}=0.\,}$

Applying the equation above and setting z=1 produces

${\displaystyle {\frac {\partial f}{\partial x}}(X,Y)\cdot x+{\frac {\partial f}{\partial y}}(X,Y)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,1)=0}$

as the equation of the tangent line.^[15] The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.^[14]

If the curve is given parametrically by

${\displaystyle x=x(t),\quad y=y(t)}$

then the slope of the tangent is

${\displaystyle {\frac {dy}{dx}}={\frac {dy}{dt}}{\bigg /}{\frac {dx}{dt}}}$

giving the equation for the tangent line at ${\displaystyle \,t=T,\,X=x(T),\,Y=y(T)}$ as^[16]

${\displaystyle {\frac {dx}{dt}}(T)\cdot (y-Y)={\frac {dy}{dt}}(T)\cdot (x-X).}$

If

${\displaystyle {\frac {dx}{dt}}(T)={\frac {dy}{dt}}(T)=0,}$

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.

Normal line to a curve

The line perpendicular to the tangent line to a curve at the point of tangency is called the normal line to the curve at that point. The slopes of perpendicular lines have product −1, so if the equation of the curve is y = f(x) then slope of the normal line is

${\displaystyle -1{\bigg /}{\frac {dy}{dx}}}$

and it follows that the equation of the normal line at (X, Y) is

${\displaystyle (x-X)+{\frac {dy}{dx}}(y-Y)=0.}$

Similarly, if the equation of the curve has the form f(x, y) = 0 then the equation of the normal line is given by^[17]

${\displaystyle {\frac {\partial f}{\partial y}}(x-X)-{\frac {\partial f}{\partial x}}(y-Y)=0.}$

If the curve is given parametrically by

${\displaystyle x=x(t),\quad y=y(t)}$

then the equation of the normal line is^[16]

${\displaystyle {\frac {dx}{dt}}(x-X)+{\frac {dy}{dt}}(y-Y)=0.}$

Angle between curves

The angle between two curves at a point where they intersect is defined as the angle between their tangent lines at that point. More specifically, two curves are said to be tangent at a point if they have the same tangent at a point, and orthogonal if their tangent lines are orthogonal.^[18]

Multiple tangents at a point

The limaçon trisectrix: a curve with two tangents at the origin.

The formulas above fail when the point is a singular point. In this case there may be two or more branches of the curve that pass through the point, each branch having its own tangent line. When the point is the origin, the equations of these lines can be found for algebraic curves by factoring the equation formed by eliminating all but the lowest degree terms from the original equation. Since any point can be made the origin by a change of variables (or by translating the curve) this gives a method for finding the tangent lines at any singular point.

For example, the equation of the limaçon trisectrix shown to the right is

${\displaystyle (x^{2}+y^{2}-2ax)^{2}=a^{2}(x^{2}+y^{2}).\,}$

Expanding this and eliminating all but terms of degree 2 gives

${\displaystyle a^{2}(3x^{2}-y^{2})=0\,}$

which, when factored, becomes

${\displaystyle y=\pm {\sqrt {3}}x.}$

So these are the equations of the two tangent lines through the origin.^[19]

When the curve is not self-crossing, the tangent at a reference point may still not be uniquely defined because the curve is not differentiable at that point although it is differentiable elsewhere. In this case the left and right derivatives are defined as the limits of the derivative as the point at which it is evaluated approaches the reference point from respectively the left (lower values) or the right (higher values). For example, the curve y = |x | is not differentiable at x = 0: its left and right derivatives have respective slopes −1 and 1; the tangents at that point with those slopes are called the left and right tangents.^[20]

Sometimes the slopes of the left and right tangent lines are equal, so the tangent lines coincide. This is true, for example, for the curve y = x ^2/3, for which both the left and right derivatives at x = 0 are infinite; both the left and right tangent lines have equation x = 0.