D[t^(-3/2) Exp[- (x^2 + y^2 + z^2)/(4 t)], 
      t]

D[t^(-3/2) Exp[- (x^2 + y^2 + z^2)/(4 t)], 
      x]

D[-(((E^(((-(x^2) - (y^2) - (z^2))/(4 t)))) x)/(2 (t^(5/2)))), x]