C4 Differentiation - Implicit differentiation                                  PhysicsAndMathsTutor.com 
               
                 
              1.     A curve C has equation 
                                                              x   2
                                                            2  + y  = 2xy 
                     Find the exact value of  dy  at the point on C with coordinates (3, 2). 
                                              dx
                                                                                                               (Total 7 marks) 
               
                 
              2.     The curve C has the equation 
                                          cos2x + cos3y = 1,        −π ≤ x ≤ π , 0 ≤ y ≤ π  
                                                                       4        4             6
                     (a)   Find  dy  in terms of x and y. 
                                  dx
                                                                                                                            (3) 
                 
                     The point P lies on C where x =  π . 
                                                       6
                     (b)   Find the value of y at P. 
                                                                                                                            (3) 
                        
                     (c)   Find the equation of the tangent to C at P, giving your answer in the form 
                           ax + by + cπ = 0, where a, b and c are integers. 
                                                                                                                            (3) 
                                                                                                               (Total 9 marks) 
               
                 
                                                      –2x         2
              3.     The curve C has the equation ye     = 2x + y . 
                     (a)   Find  dy  in terms of x and y. 
                                  dx
                                                                                                                            (5) 
              Edexcel Internal Review                                                                                         1 
              C4 Differentiation - Implicit differentiation                                  PhysicsAndMathsTutor.com 
               
               
                     The point P on C has coordinates (0, 1). 
                     (b)   Find the equation of the normal to C at P, giving your answer in the form ax + by + c = 0, 
                           where a, b and c are integers. 
                                                                                                                            (4) 
                                                                                                                (Total 9 marks) 
               
                 
                                                  2         3
              4.     A curve C has the equation y  – 3y = x  + 8. 
                     (a)   Find  dy  in terms of x and y. 
                                  dx
                                                                                                                            (4) 
                        
                     (b)   Hence find the gradient of C at the point where y = 3. 
                                                                                                                            (3) 
                                                                                                                (Total 7 marks) 
               
                 
                                              2   2
              5.     A curve has equation 3x  – y  + xy = 4. The points P and Q lie on the curve. The gradient of the 
                     tangent to the curve is  8  at P and at Q. 
                                             3
                    (a)    Use implicit differentiation to show that y – 2x = 0 at P and at Q. 
                                                                                                                            (6) 
                     (b)   Find the coordinates of P and Q. 
                                                                                                                            (3) 
                                                                                                                (Total 9 marks) 
               
                 
              6.     A curve is described by the equation 
                                                       3     2
                                                      x  – 4y  = 12xy. 
                     (a)   Find the coordinates of the two points on the curve where x = –8. 
                                                                                                                            (3) 
              Edexcel Internal Review                                                                                         2 
              C4 Differentiation - Implicit differentiation                                  PhysicsAndMathsTutor.com 
               
                     (b)   Find the gradient of the curve at each of these points. 
                                                                                                                            (6) 
                                                                                                                (Total 9 marks) 
               
                 
              7.     A set of curves is given by the equation sin x + cos y = 0.5. 
                     (a)   Use implicit differentiation to find an expression for  dy . 
                                                                                   dx
                                                                                                                            (2) 
                 
                     For –π < x < π and –π < y < π, 
                     (b)   find the coordinates of the points where  dy = 0. 
                                                                      dx
                                                                                                                            (5) 
                                                                                                                (Total 7 marks) 
               
                 
              8.     A curve C is described by the equation 
                                                          2     2  
                                                        3x – 2y    + 2x – 3y + 5 = 0. 
                     Find an equation of the normal to C at the point (0, 1), giving your answer in the form 
                     ax + by + c = 0, where a, b and c are integers. 
                                                                                                                (Total 7 marks) 
               
                 
              9.     A curve C is described by the equation 
                                                          2     2
                                                        3x  + 4y  – 2x + 6xy – 5 = 0. 
                     Find an equation of the tangent to C at the point (1, –2), giving your answer in the form 
                     ax + by + c = 0, where a, b and c are integers. 
                                                                                                                (Total 7 marks) 
               
              Edexcel Internal Review                                                                                         3 
              C4 Differentiation - Implicit differentiation                                  PhysicsAndMathsTutor.com 
               
              10.    The value £V of a car t years after the 1st January 2001 is given by the formula 
                                                                               –t
                                                            V = 10 000 × (1.5) . 
                     (a)   Find the value of the car on 1st January 2005. 
                                                                                                                            (2) 
                        
                     (b)   Find the value of  dV  when t = 4. 
                                               dt
                                                                                                                            (3) 
                        
                     (c)   Explain what the answer to part (b) represents. 
                                                                                                                            (1) 
                                                                                                                (Total 6 marks) 
               
                 
              11.    A curve has equation 
                                   2           2
                                  x  + 2xy – 3y  + 16 = 0. 
                     Find the coordinates of the points on the curve where  dy  = 0. 
                                                                             dx
                                                                                                                (Total 7 marks) 
               
                 
                                                  2           2
              12.    The curve C has equation 5x  + 2xy – 3y  + 3 = 0. The point P on the curve C has coordinates 
                     (1, 2). 
                    (a)    Find the gradient of the curve at P. 
                                                                                                                            (5) 
                        
                     (b)   Find the equation of the normal to the curve C at P, in the form y = ax + b, where a and b 
                           are constants. 
                                                                                                                            (3) 
                                                                                                                (Total 8 marks) 
               
              Edexcel Internal Review                                                                                         4