Equation of Tangent and Normal to a Curve

Once you know the slope of the tangent (or normal) at a point on a curve, writing down its equation is just an application of the familiar point-slope form: a line through $(x_1, y_1)$ with slope $m$ satisfies $(y - y_1) = m(x - x_1)$.

Let $y = f(x)$ be a curve and $A(x_0, y_0)$ a point lying on it. Substituting the tangent's slope from the previous section as $m$ gives the equation of the tangent at $A$:

and substituting the normal's slope in the same way gives the equation of the normal at $A$: