Partial Derivatives

f / x: differentiate in x, hold y constant.

The idea

A partial derivative is the rate of change of a function of several variables when only one of its inputs varies. A function of one variable has a single derivative at each point; a function $f(x, y)$ changes at its own rate in every direction, and the simplest directions to measure are along the axes.

Fix $y$ at some value. What remains is an ordinary function of the single variable $x$, and it has an ordinary derivative. That derivative is the partial derivative of $f$ with respect to $x$: $\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x + h,\, y) - f(x, y)}{h}.$ We read it aloud as the partial derivative of $f$ with respect to $x$, and also write it $f_x$ or $\partial_x f$; the symbol $\partial$ in place of $d$ announces that another variable is being held fixed. Exchanging the roles of $x$ and $y$ defines $\partial f / \partial y$ the same way. On the graph of $f$ — the surface $z = f(x, y)$ — holding one variable fixed slices the surface in a curve, and the partial derivative is the slope of that slice's tangent.

Every rule from single-variable calculus — product, quotient, chain — applies word for word, because once the other variable is held fixed there is only one variable left to differentiate.

Ways to work on it

• Walkthrough. Compute the partial derivatives f_x and f_y of a two-variable function.
• Practice. Evaluate f_x and f_y at a point for a random small f(x, y).
• Hardest. Compute the mixed second partial f_xy at a point and compare it with f_yx.

Not sure where to start? Take the ten-question placement test.