Subtangent

The subtangent is a term from analytical geometry . We consider a point on a differentiable curve (red) and form the tangent (green). The projection of the tangent onto the abscissa is then called the subtangent (yellow in the picture).

If a curve is differentiable at one point , the subtangent is the distance between the point on the abscissa and the zero point of the tangent. ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$

The zero and thus the subtangent can be obtained via the equation of the tangent ${\displaystyle t(x)=f'(x_{0})\cdot (x-x_{0})+f(x_{0})}$ ${\displaystyle x_{s}={\frac {-f(x_{0})}{f'(x_{0})}}+x_{0}}$ ${\displaystyle \left|{\frac {f(x_{0})}{f'(x_{0})}}\right|.}$

The amount of the subtangent of the function is constant 1 for all tangents, since: ${\displaystyle e}$ ${\displaystyle f(x)=e^{x}}$

${\displaystyle \left|{\frac {e^{x_{0}}}{e^{x_{0}}}}\right|=1}$

As an analogy to the normal there are the subnormal .