Exercise 12 Let f (x, y) be a differentiable function. Prove that V(f ( x, y)g (x, y) ) = f ( x, y) Vg(x, y) + Vf(x, y)g(x,y). Here is a "mini lesson" on proving two things are equal: 1. Start with the left-hand side and rewrite it using relevant definitions (in this case the gradient). 2 If Step 1 doesn't lead you straight to the thing on the right-hand side, then start with the right-hand side and rewrite it using relevant definitions. With hope, the work you do in rewriting the right-hand side equals something you obtained while working on the left-hand side. They meet in the middle. 3. Hopefully, Steps 1 and 2 enable you to see how the two sides are equal. Starting with just one of the sides, show through a sequence of equalities that the left-hand side equals the right-hand side. The final step in your proof should be the equality. Never start a proof with the statement to be proved