relation between the demand function and the marginal utility function Can one say that the demand curve equals the marginal utility function? I have one course where this is stated.

However, the demand curve dan be derived from combining the utility function with the budget curve, while varying the price of the one good of interest and plotting the corresponding quantities found from the equilibria in the goods space and the budget curve.

While I do understand this method, I don't understand how the demand curve can be seen as the marginal utility function using this method (or any unknown other).

Please enlighten me :).
This is true for quasilinear utility function. Suppose
$$U = d + V(x),
$$
where $d$ is the numeraire. The budget constraint is given by $p_x x + d = I$. The utility is then
$$U = I - p_x x + V(x).$$
Maximizing this, we have the following FOC: $p_x = V'(x)$. That is, for a given $x$, my willingness to pay is $V'(x)$, the marginal utility.