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Unit 9: Differential Calculus




                                                                                                Notes
                             1
               Example: Solve   3
                            x
                   d   1    d   3
          Solution:    3       x
                  dx x     dx
                         3 x  3 1  3x  4
                            1
                         3   4
                           x
                        3
                       x 4

                                   2
                             2
               Example: Solve (x  + 1) (x  – 1)
                   d    2     2       d   4
          Solution:    x  1 x    1       x   1  (on multiplication)
                  dx                  dx
                               d   4   d
                                  x       1
                               dx     dx
                        3       3
                      4x   0  4x

                            x
               Example: Solve e (x – 5)logx
                   d   x
          Solution:   e x  5 log x
                  dx
                         d         x      d                  d   x
                  x
                 e  x  5    logx   e  logx   x  5   x  5 logx    e
                         dx              dx                  dx
                         1   x                    x
                  x
                 e  x  5    e  log x  1  x  5 log x e
                         x
                       5
                  x
                 e  1     log x x log x  5log x
                       x
                  x    5
                 e  1     log x x log x  5log x
                       x
                       5
                  x
                 e  1     4log x x log x
                       x

                               e x log x
                         Solve     2
              Task             1 x
                           dy        2x  1
               Example: Find   ,  if, y  2
                           dx        1 x
          Solution:
                          2 d                 d     2
                      1 x      2x  1   2x  1     1 x
                 dy         dx               dx
                 dx                1 x 2  2


                    1 x 2  2   2x  1   2x
                                 2
                                2
                            1 x



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