In Total and Marginal Utility marginal utility was the change in total utility from one more unit, read off a table. With a smooth utility function we can do better and find the rate at an exact bundle. The tool is the partial derivative from Partial Derivatives: change one good a little, hold the other fixed, and see how fast $U$ rises.
Key idea
$$MU_x = \frac{\partial U}{\partial x}, \qquad MU_y = \frac{\partial U}{\partial y}.$$ $MU_x$ is the extra utility per unit of a small increase in $x$, with $y$ held where it is. A small change $\Delta x$ raises utility by about $MU_x\,\Delta x$.
Why it works
The table version is a difference quotient, $\Delta U/\Delta x$ with $\Delta x = 1$. Shrink the step and the quotient settles on a limit: $\displaystyle \lim_{h \to 0} \frac{U(x + h, y) - U(x, y)}{h}$. That limit is the partial derivative. Because $y$ stays fixed, you differentiate as if $y$ were a number: a factor of $y$ rides along, and a term with only $y$ in it drops out.
Worked example: A Cobb-Douglas utility
$U(x, y) = x^{1/2}y^{1/2}$. Find $MU_x$ and $MU_y$ at $(4, 9)$.
- Treat $y$ as a constant: $MU_x = \tfrac{1}{2}x^{-1/2}y^{1/2} = \dfrac{\sqrt{y}}{2\sqrt{x}}$.
- Treat $x$ as a constant: $MU_y = \tfrac{1}{2}x^{1/2}y^{-1/2} = \dfrac{\sqrt{x}}{2\sqrt{y}}$.
- At $(4, 9)$: $MU_x = \dfrac{3}{2 \cdot 2} = \dfrac{3}{4}$ and $MU_y = \dfrac{2}{2 \cdot 3} = \dfrac{1}{3}$.
- So a little more $x$ (say 0.1 unit) adds about $0.075$ to utility, and a little more $y$ adds about $0.033$.
Answer: $MU_x = 3/4$, $MU_y = 1/3$.
Worked example: A product with a shift
$U(x, y) = (x + 2)y$. Find $MU_x$ and $MU_y$ at $(3, 5)$.
- $MU_x = y$ (the factor $y$ rides along; the derivative of $x + 2$ is 1).
- $MU_y = x + 2$.
- At $(3, 5)$: $MU_x = 5$ and $MU_y = 5$.
Answer: $MU_x = MU_y = 5$.
Common mistake
Differentiating with respect to both goods at once, or setting the other good to zero. For $U = x^{1/2}y^{1/2}$, setting $y = 0$ makes $U = 0$ for every $x$, so you would wrongly find $MU_x = 0$. The other good is held fixed at its actual value, not removed. Check: your $MU_x$ should usually still contain $y$, and plugging in the bundle should give a positive number when more of $x$ helps.
Diminishing marginal utility, $\partial^2 U/\partial x^2 < 0$, is a property of the particular utility function, not of the preferences. $U = xy$ has constant $MU_x = y$, and $V = (xy)^2$ has rising $MU_x = 2xy^2$, yet both describe the same person. What survives every increasing transformation is the ratio of marginal utilities, the subject of The Marginal Rate of Substitution.