The first derivative (f'(x)) is the gradient of the tangent to the graph of (y=f(x)) at a general point ((x,y)).
[b]uEquation of tangents[/u]/b
Using the equation of straight lines (y-y1=m(x-x1) ), the equation of a tangent to graph at ((x_1,y_1)) can be found by using the gradient (m = f'(x_1)) and the values of the (x_1) and (y_1) coordinates.
[b]uEquation of normals[/u]/b
Similar to the above, except the gradient of the normal is the negative reciprocal of the gradient of the tangent, i.e. (m = \dfrac{-1}{f'(x1)} ).
[b]uEquation of tangents[/u]/b
(y-y1=f'(x1)(x-x1) )
[b]uEquation of normals[/u]/b
(y-y1=\dfrac{-1}{f'(x1)}(x-x1) )