Consider a curve whose equation is y = f(x). On this curve, take a point P(x1,y1).
Assuming that the tangent at this point is not parallel to the coordinate axes, the equation of the tangent line at P is the equation of a straight line with slope gradient m passing through (x1,y1).
y – y1 = m(x – x1)
The slope m = f′(x1) = dy/dx at (x1,y1)
The equation of the tangent is of the form
y – y1 = f′(x1)(x – x1)
The normal to a curve at a given point is a straight line passing through the given point, perpendicular to the tangent at this point.
The slope of the normal m′ and that of the tangent m are connected by the equation
m′ = – 1/m
The equation of a normal to a curve y = f(x) at a point P(x1,y1) is
y – y1 = m′(x – x1)