Question

Define frs,t)-sret and find firss, first and fss rss Jrst, an sst

Answer 1

Let y = f(r, s, t) = sre^rt

Then

f_s = $\partial$ y/ $\partial$ s = re^rt

f_t = $\partial$ y/ $\partial$ t = sr²e^rt

f_ss = $\partial$ / $\partial$ s( $\partial$ y/ $\partial$ s) = $\partial$ / $\partial$ s(re^rt) = 0

f_st = $\partial$ / $\partial$ s( $\partial$ y/ $\partial$ t) = $\partial$ / $\partial$ s(sr²e^rt) = r²e^rt

f_rss = $\partial$ / $\partial$ r(f_ss) = $\partial$ / $\partial$ r(0) = 0

f_rst = $\partial$ / $\partial$ r(f_st) = $\partial$ / $\partial$ r(r²e^rt) = (r² x r x e^rt) + (e^rt x 2r) = (r³ x e^rt) + (e^rt x 2r)

f_sst = $\partial$ / $\partial$ s(f_st) = $\partial$ / $\partial$ s(r²e^rt) = 0