Solutions

In[48]:=

(*1*)Dt[x y + z^2 x + y z^3]//Simplify

Out[48]=

(y + z^2) Dt[x] + (x + z^3) Dt[y] + z (2 x + 3 y z) Dt[z]

The greek alphabet can be written as "Esc-theta-Esc", etc. α β γ δ

In[49]:=

(*2*)Dt[1/r^2 * Sin[θ] Cos[ϕ]]//Simplify

Out[49]=

(r Cos[θ] Cos[ϕ] Dt[θ] - Sin[θ] (2 Cos[ϕ] Dt[r] + r Dt[ϕ] Sin[ϕ]))/r^3

Dt automatically assumes that all symbols are a function of the variable with respect to which the total derivative is calculated. You can override this assumption with Constants->{...}.

In[50]:=

(*3*)Dt[Sqrt[x^2 + y^2], t]

Out[50]=

(2 x Dt[x, t] + 2 y Dt[y, t])/(2 (x^2 + y^2)^(1/2))

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