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 CurveOnce the gradient of a tangent to a curve has been found, the formula y − y_{1} = m(x − x_{1}) can be used to find the equation of the tangent at the point (x_{1}, y_{1}). Example There are two methods for solving this problem.
Finding the Equation of the Normal to a CurveThe 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 m_{1} is perpendicular to another line with gradient m_{2} then m_{1} x m_{2} = 1. ⇒ m_{1} = 1/ m_{2} Example Find the equation of the normal to the curve x^{2} + y^{2} = 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:
