For each of the following functions show that the second-order mixed partial derivatives are equal (i.e., the order of differentiation doesn’t matter). If you need something extra to do, check the same thing for the third-order mixed partial derivatives.

You should probably use a computer to do the third-order mixed partial derivatives, just to save time.

a. f

(a,

b) = a 2 + b 2

.

b. g(r, t) = 1 + r 2

t + t 3

r.

c. X(u, v) = sin(uv).

d. N(x, y, w) = e xyw.

e. A(q, p) = ln(qp).

f. C(x, y, z) = cos(