Let P (x, y) be any point on y = f (x). Let the tangent drawn at ‘ P ‘ meets the x-axis at ‘ T ‘, and normal drawn at ‘ P ‘ meets the x-axis at ‘ N ‘.

PT is called length of the tangent and PN is called the length of the normal.

If ‘ P_{1} ‘ be the projection of the point P on the x-axis then TP_{1} is called the sub-tangent (projection of line segment PT on the x-axis) and NP_{1} is called the sub normal (projection of line segment PN on the x-axis).

Let ∠PTN = θ => ∠P_{1}PN = θ

We have tanθ = dy/dx and PP_{1} = |y|

Now, PT= |y cosec θ|

or, PT =

Hence length of the tangent PT =

Now, PN = |y sec θ| =

Length of the normal, PN =

Now, TP_{1} = |y cot θ| =

=> Sub-tangent TP_{1} =

Finally, NP_{1} = |y tan θ| =

=> Sub-normal, NP_{1} =