Kenyon College                                                              paquind@kenyon.edu
                                                             Math 333
                                      Some Practice with Partial Derivatives
                     Suppose that f(t,y) is a function of both t and y. The partial derivative of f with
                     respect to y, written
                                                                   ∂f,
                                                                   ∂y
                     is the derivative of f with respect to y with t held constant. To find ∂f, you should
                                                                                                     ∂y
                     consider t as a constant and then find the derivative of f with respect to y.
                                                        2      3
                     Example. Suppose f(t,y) = t sin(y ). Then
                                                         ∂f      2       3     2
                                                         ∂y =t cos(y )·3y .
                     Some Practice Problems.
                                                3 2         ∂f
                       1. Suppose f(t,y) = t y . Find ∂y.
                                                 t+y        ∂f
                       2. Suppose f(t,y) = e        . Find ∂y.
                                                    2           ∂f
                       3. Suppose f(t,y) = ln(t y). Find ∂y.
                       4. Suppose f(t,y) = cos(ty). Find ∂f.
                                                                ∂y
                       5. Suppose f(t,y) =         ty    . Find ∂f.
                                                    3  2
                                                sin(t +y )       ∂y
                     Answers to the Practice Problems.
                           ∂f      3
                       1. ∂y = 2t y
                           ∂f     t+y
                       2. ∂y = e
                           ∂f     1    2
                       3.     = 2 ·t
                           ∂y    t y
                       4. ∂f = −sin(ty)·(t)
                           ∂y
                                       3   2
                       5. ∂f = t−cos(t +y )2y
                           ∂y        2 3   2
                                   sin (t +y )
                     Math 333: Diff Eq                               1                              Partial Derivatives