Tangents and Normals:

The equation of a straight line passing through a given point (x 1 , y1 ) having finite slope m is given by

 y – y1 = m (x – x1 )

The slope of the tangent to the curve y = f(x) at the point (x1 , y1 ) is given by

So the equation of the tangent at (x1 , y1 ) to the curve y = f (x) is given by

 y – y0 = f ′(x1 )(x – x1 )

Note: If a tangent line to the curve y = f (x) makes an angle θ with x-axis in the positive direction, then slope of the tangent =dy/ dx = tan θ .

Particular cases:

[1] If slope of the tangent line is zero, then tan θ = 0 and so θ = 0 which means the tangent line is parallel to the x-axis. In this case, the equation of the tangent at the point (x0 , y0 ) is given by y = y0 .

Example ;

 Find the slope of the tangent to the curve y = x 3 – x at x = 2.

Solution :

The slope of the tangent at x = 2 is given by