Quotient rule
Dt[f[x]/g[x],x]==
Dt[f[x],x]/g[x]+f[x]Dt[1/g[x],x]==
f'[x]/g[x]+f[x]Dt[1/g[x]]/Dt[g[x]]Dt[g[x],x]==
f'[x]/g[x]-f[x]/g[x]^2g'[x]==
(f'[x]g[x]-f[x]g'[x])/g[x]^2

Dt[x^a, x] ==
Dt[E^Log[x^a], x] ==
Defer[Dt[E^Log[x^a]]/Dt[Log[x^a]]]*Dt[a*Log[x], x] ==
(a*E^(a*Log[x]))/x ==
(a*x^a)/x ==
a*x^(a - 1)

Dt[Log[a, x], x] ==
Limit[(Log[a, t + x] - Log[a, x])/t, t -> 0] ==
Limit[Log[a, (t + x)/x]/t, t -> 0] ==
Limit[Log[a, t/x + 1]/t, t -> 0] ==
Limit[x/t*Log[a, t/x + 1]/x, t -> 0] ==
Limit[Log[a, (t/x + 1)^(x/t)]/x, t -> 0] ==
Log[a, Limit[(t/x + 1)^(x/t), t -> 0]]/x ==
Log[a, e]/x ==
1/(x*Log[a])

Dt[a^x, x] ==
Limit[(a^(t + x) - a^x)/t, t -> 0] ==
a^x*Limit[(a^t - 1)/t, t -> 0] ==
a^x*Limit[(a^t - 1)/Log[a, (a^t - 1) + 1], t -> 0] ==
a^x/Limit[Log[a, (a^t - 1) + 1]/(a^t - 1), t -> 0] ==
a^x/Limit[Log[a, ((a^t - 1) + 1)^(1/(a^t - 1))], t -> 0] ==
a^x/Log[a, Limit[(b + 1)^(1/b), b -> 0]] ==
a^x/Log[a, E] ==
a^x*Log[a]

Dt[Sin[x], x] ==
Limit[(Sin[t + x] - Sin[x])/t, t -> 0] ==
Limit[(2*Sin[t/2]*Cos[t/2 + x])/t, t -> 0] ==
Limit[(Sin[t/2]*Cos[t/2 + x])/(t/2), t -> 0] ==
Limit[Cos[t/2 + x], t -> 0] ==
Cos[x]

Dt[Cos[x], x] ==
Limit[(Cos[t + x] - Cos[x])/t, t -> 0] ==
Limit[-((2*Sin[t/2]*Sin[t/2 + x])/t), t -> 0] ==
-Limit[(Sin[t/2]*Sin[t/2 + x])/(t/2), t -> 0] ==
-Limit[Sin[t/2 + x], t -> 0] ==
-Sin[x]