a. Practically speaking, you compute the differential in much the same way you compute a derivative via implicit differentiation, but you omit the variable with respect to which you are differentiating.

Aside: Compare this to what happens when you differentiate both sides with respect to some other independent parameter, say
:

b. This is just a matter of plugging in
and
.

2x-4y=8
2x=4y+8
x=2y+4
^^^^^^^
Let m and h represent hours Mai spends mowing and hauling, respectively. Then (m+h) will be the number of hours Priya spends babysitting. In order for their earnings to be equal, we must have
7m +14h = 8.40(m+h)
Subtract 7m+8.40h: 5.60h = 1.40m
Divide by 1.40: m = 4h
Then the total number of hours worked by either person is
m + h = (4h) +h = 5h
When only whole numbers of hours are worked, the smallest number of hours that will make earnings equal is 5h, with h=1, or 5 hours. In that time, each will earn 5×$8.40 = $42.
Therefore, each of them must work 5 hours and earn $42 before they go to the movies and Mai will work 4 hours mowing and 1 hour hauling.