Tangent and Normal Formulas

The formulas of tangent and normal to any curve at a given point are listed below.

  1. {\left. {\frac{{dy}}{{dx}}} \right|_p} is the slope of the tangent to the curve y = f\left( x \right) at the point p
  2. In a plane curve r = f\left( \theta \right),

    \tan \phi = r\frac{{d\theta }}{{dr}}

  3. The equation of the tangent at a point P\left({{x_1},{y_1}} \right) is

    \left({y - {y_1}} \right) = {\left. {\frac{{dy}}{{dx}}} \right|_p}\left( {x - {x_1}} \right)

  4. The equation of the normal at a point P\left({{x_1},{y_1}} \right) is

    \left({x - {x_1}} \right) = {\left. {\frac{{dy}}{{dx}}} \right|_p}\left( {y - {y_1}} \right)

  5. Consider that a curve c is defined by y = f\left( x \right), and p is the length of the perpendicular from O\left( {0,0} \right) to the tangent at the point \left({{x_1},{y_1}} \right) of the curve. Then

    p = \frac{{\left| {{y_1} - {x_1}\frac{{dy}}{{dx}}} \right|}}{{\sqrt {1 - {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} }}