(*f[k] denotes the partial derivative of u[k] with respect to x*)
(*g[1] denotes the partial derivative of u[1] with respect to t*)
f[1] = u[2];
f[2] = g[1];

trans = {u[1] -> w[1, 1], u[2] -> w[1, 1] w[1, 2]};

(*Denote y[1,1] as the partial derivative of w[1,1] with respect to x*)
z[1, 1] = f[1] /. trans

(*Denote y[1,2] as the partial derivative of w[1,2] with respect to x*)
(*Denote z[1,1] as the partial derivative of w[1,1] with respect to t*)
z[1, 2] = Factor[(u[2]^2 - u[1] g[1])/u[2]^2 /. trans] /. g[1]->g[1,1]

D[u[1][x, t], x] = u[2][x, t];

Set::write: Tag D in !(*SubscriptBox[([PartialD]), (x)]((u[1])[x, t])) is Protected. >>