Tangents and Normals: Derivative As Measure Of Slope

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)