#### Problem 39E

39. Establish the following rules for working with differentials (where $c$ denotes a constant and $u$ and $v$ are functions of $x$ ).

(a) $d c=0$

(b) $d(c u)=c d u$

(c) $d(u+v)=d u+d v$

(d) $d(u v)=u d v+v d u$

(e) $d\left(\frac{u}{v}\right)=\frac{v d u-u d v}{v^{2}}$

(f) $d\left(x^{n}\right)=n x^{n-1} d x$