Answer
See proofs
Work Step by Step
a)
$dc$ = $\frac{dc}{dx}dx$ = $0dx$ = $0$
b)
$d(cu)$ = $\frac{d}{dx}(cu)dx$ = $c\frac{du}{dx}dx$ = $cdu$
c)
$d(u+v)$ = $\frac{d}{dx}(u+v)$ = $\left(\frac{du}{dx}+\frac{dv}{dx}\right)dx$ = $\frac{du}{dx}dx+\frac{dv}{dx}dx$ = $du+dv$
d)
$d(uv)$ = $\frac{d}{dx}(uv)dx$ = $\left(u\frac{dv}{dx}+v\frac{du}{dx}\right)dx$ = $u\frac{dv}{dx}dx+v\frac{du}{dx}dx$ = $udv+vdu$
e)
$d\left(\frac{u}{v}\right)$ = $\frac{d}{dx}\left(\frac{u}{v}\right)dx$ = $\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}dx$ = $\frac{v\frac{du}{dx}dx-u\frac{dv}{dx}dx}{v^{2}}$ = $\frac{vdu-udv}{v^{2}}$
f)
$d(x^{n})$ = $\frac{d}{dx}(x^{n})dx$ = $nx^{n-1}dx$
