The document discusses finding the gradient of implicitly defined curves using implicit differentiation. It provides the example curve x^3 + y^3 + x^2 - y = 0, which cannot be written explicitly as y = f(x) or x = f(y). To find the gradient at any point on an implicitly defined curve, the technique of implicit differentiation is used. The document then poses the question of finding the equation of the tangent line to the implicitly defined circle x^2 + y^2 - 6x + 2y - 3 = 0 at the point (1, 2).

1. Starter

2. Objective: to find the gradients of implicit curves

3. An equation of a curve is implicit if it is not of the form y = f( x ) or x = f( y ). The implicit curve x ³ + y ³ + x ² – y = 0 cannot be rearranged into an explicit form. In order to find the gradient at any point on the curve, we use implicit differentiation. Curves defined implicitly

4. Implicit Differentiation Returning to the Starter questions:

5. Question Find the equation of the tangent at the point (1 , 2) to the circle x ² + y ² – 6 x + 2 y – 3 = 0 .

6. Core 3 & 4 Textbook Exercise 8A Page 333 Questions 2 – 8