By differentiating and finding the derived or gradient function we obtain an equation which will give the gradient at any point on the curve. The gradient for a particular value of x can then be obtained by substitution.
Finding the Equation of the Tangent to a Curve
Once the gradient of a tangent to a curve has been found, the formula y − y1 = m(x − x1) can be used to find the equation of the tangent at the point (x1, y1).
There are two methods for solving this problem.
Finding the Equation of the Normal to a Curve
The normal to a tangent to a curve at a point is the line perpendicular to the tangent at the point of contact.
If a line with gradient m1 is perpendicular to another line with gradient m2 then m1 x m2 = -1. ⇒ m1 = -1/ m2
Find the equation of the normal to the curve x2 + y2 = 4 at the point (√2, √2)
The two methods shown above can be used to find that the gradient of the tangent at (√2, √2) is m = -1
The gradient of the normal is m = -1/ -1 = 1
Hence the equation is: