Editing Tangent (section)

===Equations===
When the curve is given by ''y'' = ''f''(''x'') then the slope of the tangent is <math>dy/dx,</math>
so by the [[Linear equation#Point–slope form or Point-gradient form|point–slope formula]] the equation of the tangent line at (''X'',&nbsp;''Y'') is

:<math>y-Y=\frac{dy}{dx}(X) \cdot (x-X)</math>

where (''x'',&nbsp;''y'') are the coordinates of any point on the tangent line, and where the derivative is evaluated at <math>x=X</math>.<ref name=E191>Edwards Art. 191</ref>

When the curve is given by ''y'' = ''f''(''x''), the tangent line's equation can also be found<ref>Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs", ''[[Mathematical Gazette]]'', November 2005, 466–467.</ref> by using [[polynomial division]] to divide <math>f \, (x)</math> by <math>(x-X)^2</math>; if the remainder is denoted by <math>g(x)</math>, then the equation of the tangent line is given by

:<math>y=g(x).</math>

When the equation of the curve is given in the form  ''f''(''x'',&nbsp;''y'') = 0 then the value of the slope can be found by [[Implicit and explicit functions#Implicit differentiation|implicit differentiation]], giving

:<math>
\frac{dy}{dx}=-\frac{\partial f}{\partial x} \bigg/
\frac{\partial f}{\partial y}.
</math>

The equation of the tangent line at a point (''X'',''Y'') such that ''f''(''X'',''Y'') = 0 is then<ref name=E191/>

:<math>
\frac{\partial f}{\partial x}(X,Y) \cdot (x-X) +
\frac{\partial f}{\partial y}(X,Y) \cdot (y-Y) = 0.
</math>

This equation remains true if

:<math>\frac{\partial f}{\partial y}(X,Y) = 0,\quad \frac{\partial f}{\partial x}(X,Y) \neq 0,</math>

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

:<math>
\frac{\partial f}{\partial y}(X,Y)
= \frac{\partial f}{\partial x}(X,Y) = 0,
</math>

the tangent line is not defined and the point (''X'',''Y'') is said to be [[singular point of a curve|singular]].

{{clear}}
For [[algebraic curve]]s, computations may be simplified somewhat by converting to [[homogeneous coordinate]]s. Specifically, let the homogeneous equation of the curve be ''g''(''x'',&nbsp;''y'',&nbsp;''z'') = 0 where ''g'' is a homogeneous function of degree ''n''. Then, if (''X'',&nbsp;''Y'',&nbsp;''Z'') lies on the curve, [[Homogeneous function#Positive homogeneity|Euler's theorem]] implies
<math display=block>\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.</math>
It follows that the homogeneous equation of the tangent line is

:<math>
\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.
</math>

The equation of the tangent line in Cartesian coordinates can be found by setting ''z''=1 in this equation.<ref name=E192>Edwards Art. 192</ref>

To apply this to algebraic curves, write ''f''(''x'',&nbsp;''y'') as

:<math>f=u_n+u_{n-1}+\dots+u_1+u_0\,</math>

where each ''u''<sub>''r''</sub> is the sum of all terms of degree ''r''. The homogeneous equation of the curve is then

:<math>g=u_n+u_{n-1}z+\dots+u_1 z^{n-1}+u_0 z^n=0.\,</math>

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

:<math>\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</math>

as the equation of the tangent line.<ref name=E193>Edwards Art. 193</ref> The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.<ref name=E192 />

If the curve is given [[Parametric equation|parametrically]] by

:<math>x=x(t),\quad y=y(t)</math>

then the slope of the tangent is

:<math>
\frac{dy}{dx} = \frac{dy}{dt} \bigg/ \frac{dx}{dt}
</math>

giving the equation for the tangent line at <math>\, t=T, \, X=x(T), \, Y=y(T)</math> as<ref name=E196>Edwards Art. 196</ref>

:<math>\frac{dx}{dt}(T) \cdot (y-Y)=\frac{dy}{dt}(T) \cdot (x-X).</math>

If 

:<math>\frac{dx}{dt}(T)= \frac{dy}{dt}(T) =0, </math>

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.