Higher-Order Partial Derivatives

Differentiate again to climb the order

Mixed partials usually commute

What second derivatives measure

Higher orders follow the same pattern

For $f(x, y) = x^2 y$, compute $f_{xy}$ and evaluate it at the point $(2, 3)$.

Under what condition are $f_{xy}$ and $f_{yx}$ guaranteed equal at a point?

For $f(x, y) = x^2 y^3$, which pair gives $f_{yy}$ and its value at $(1, 1)$?

For $f(x, y) = x^3 y^4$, compute $f_{xy}$, the mixed partial with the $x$-differentiation first, and give its value at the point $(1, 1)$, as a plain number.

For $f(x, y) = xy$, give the value of the mixed partial $f_{xy}$, as a plain number.