When you find the partial derivative of a function with respect to a particular variable (say, x), then you treat every other variable like it's a constant. You then perform the derivative exactly as before. It takes a few problems before it "clicks", but the process will soon feel exactly the same as regular differentiation.
We have f(x, y, z, t) = (xy)/(t + 2z).
We'll first find (delf)/(delx), which can be more conveniently notated f_x. Both notations refer to the first partial derivative of f with respect to x. For f_x, we treat x like a variable and everything else like a regular number. Thus, f = (y/(t+2z))(x) and the leftmost term is considered constant. Because the derivative of the function Cx is C, where C is constant, it follows that f_x = y / (t + 2z).
A similar procedure is followed to find f_y. Since f = (x/(t + 2z))(y), it follows that f_y = x / (t+2z).
f_z is found by seeing that f = (xy)(t + 2z)^(-1). (Remember, x, y, t are constant.) Thus f_z = (-1)(xy)(t + 2z)^(-2)(2). Note that we used the chain rule. This can be simplified as f_z = (-2xy)/(t + 2z)^2.
Just as we found f_z, f_t = (-1)(xy)(t + 2z)^(-2). This can be written f_t = (-xy)/(t+2z)^2.
If you still find the process difficult, simply rewriting the equation with capital letters except in the variable you're differentiating with respect to, can help. For example f_z is more easily calculated if you write f = (XY)/(T + 2z). Mentally, it is more clear that x, y, t are constants.
