Page 112 - Foundations Of Differential Calculus

5. On the Differentiation of Algebraic Functions of One Variable 95
tion, namely,

y = p ± q ± r ± s,

then by this rule we will have
dy = dp ± dq ± dr ± ds,

which is clear from the rule stated previously.

172. If the parts are joined by multiplication, so that
y = pq,

it is clear that if we suppose that only the part p is variable, then the
differential will be equal to qdp. If the other part q is the only variable,
then the differential is equal to pdq. When we add these two differentials
we obtain the desired differential
dy = qdp + pdq,

just as we proposed above. If there are several parts joined by multiplica-
tion, for example,

y = pqrs,

and we successively let each part be variable, we obtain the differentials
qrs dp, prs dq, pqs dr, pqr ds,

whose sum gives the desired differential

dy = qrs dp + prs dq + pqs dr + pqr ds,
as we have already seen. Therefore, the differential is obtained from the
differentials of all of the parts, whether they are joined by addition, sub-
traction, or multiplication.

173. If the parts of the function are joined by division, for example,
p
y = ,
q
according to the rule we first let p be variable, and since q is constant,
the differential is equal to dp/q. Next we let q alone be variable, and since
2
y = pq −1 , the differential is equal to −pdq/q . When we join the two
differentials we have the differential of the given function
dp pdq qdp − pdq
dy = − = ,
q q 2 q 2