Tangent of a curve
1.  Tangent: Means touching line
2.  Tangent: Let y  =  f (x) be a curve and P be a point on the curve.
           If Q (P) is a point on the curve, then is called a secant line. If the secant line  approaches the same limiting position as Q moves along the curve and approaches to P from either side, then limiting position is called a tangent line to the curve at P. The point P is called point of contact of the tangent line to the curve y = f(x).
Note: If P is a point on the curve y = f(x) then there exists all most one tangent at P to the curve y = f(x).
Normal of a curve
Normal: Let y = f(x) be a curve and P be a point on the curve. The line passing through P and perpendicular to the tangent y = f(x) at P is called the normal to the curve y = f(x) at P.
Gradient of a curve: Let y = f(x) be a curve and P be a point on the curve. The slope of the tangent to the curve y = f(x) at P is called gradient to the curve at P.

Angle between two curves
*  The acute angle which two curves intersect at a point is defined as the angle between their tangents at the point of intersection.
*  If the angle at a point of intersection is a right angle then the two curves are said to cut each other orthogonally at the point of intersection.
*  If the angle at a point of intersection is zero then the two curves are said to touch each other at the point of intersection.
*  If m1 is the slope of the tangent to the curve f(x) and m2 is the slope of the tangent to the

             curve g(x) such that 1 + m₁ m₂ 0 then tan  
*  If 1 + m₁m₂ = 0 m₁m₂ =  -1 then the two curves cut each other orthogonal at P.
*  If m₁ = m₂ then the two curves touch each other at P. In this case the two curves have a common tangent and a common normal at P.
*   Let us consider a curve whose equation is y =  f(x).
*   Let P (x₁, y₁) be a point on the curve.

 Let P (x1, y1) be a point on the curve.
*  Let the tangent at P be not perpendicular to X -axis.Let the tangent meet the X -axis in T.
*  Draw PN perpendicular to the tangent at P to meet  in N. Draw   
*  The length of the segment PT is called the length of the tangent at P.
*  The length of the segment PN is called the length of the normal at P.
*  TQ, The projection of PT on OX, is called the length of the sub tangent.
*  QN, The projection PN on OX, is called the length of the sub normal.