Multivariate Calculus - Bruce E. Shapiro

62 LECTURE 9. THE PARTIAL DERIVATIVE
Figure 9.2: The function T (x, y) discused in example 9.4.
Higher order partial derivatives
The second and higher order partial derivatives
f xx = ∂ ∂f
∂x ∂x = ∂2 f
∂x 2
f yy = ∂ ∂y
∂f
∂y = ∂2 f
∂y 2
In addition, there are mixed partial derivatives for higher orders.
f xy = (f x ) y = ∂ ∂y
∂f
∂x = ∂2 f
∂y∂x
f yx = (f y ) x = ∂ ∂f
∂x ∂y = ∂2 f
∂x∂y
Example 9.5 Find all the second order partial derivatives of
From equations (9.3) and (9.4),
f(x, y) = x 3 y 2 − 5x + 7y 3
∂f
∂x = 3x2 y 2 − 5,
∂f
∂y = 2x3 y + 21y 2
Hence
f xx = ∂ ∂f
∂x ∂x = ∂
∂x (3x2 y 2 − 5) = 6xy 2
f xy = ∂ ∂f
∂y ∂x = ∂ ∂y (3x2 y 2 − 5) = 6x 2 y
Revised December 6, 2006. Math 250, Fall 2006