Partial derivatives

Learn what a function of two variables is and how to evaluate one at an ordered pair. Then find the domain as a region of the $xy$-plane, and decide whether its boundary curve belongs to the domain.

Learn how to picture a function of two variables as a surface in three dimensions, test whether a given point lies on that surface, and fix one input to cut the surface into a familiar one-variable curve.

Learn how a horizontal plane cuts a surface into a level curve, and how to find that curve by solving $f(x,y)=c$. Then read a contour plot: which way is uphill, where the surface is steep or shallow, and where its peaks and basins lie.

Learn what a partial derivative is: the rate at which a function of multiple variables changes as one input moves and the others are held fixed. Then read the signs, zeros and relative sizes of partial derivatives straight off a contour plot.

Learn how to compute a partial derivative: evaluate one at a point by fixing the held input first, then produce it as a formula by treating the other variable as a constant. The same method extends to functions of three or more inputs, where every variable except the one named is held fixed.

Learn when a partial derivative needs the product rule and how to apply it. Then differentiate composite functions where the variable held constant enters through the inner derivative.

Learn how to read the four second-order partial derivatives of a function of two variables in both notations, and understand what each one measures: a repeated variable gives the concavity of a cross-section, and a mixed partial gives how the slope in one direction changes as the other variable increases.

Learn how to compute the second-order partial derivatives of a function of several variables by differentiating a first partial derivative once more, and understand why the Mixed Derivative Theorem lets a mixed partial be computed in whichever order is less work.