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