Question

Question 15 Find 331 - for f(x, y) = pay at (x, y) = (1,2). მცმე-2 Oe? None of the above or below 3e? 8e2 4e 2e3

Answer 1

GIven: $f(x,y)=e^{xy}$

$\frac{\partial ^3}{\partial \:y\partial \:x^2}\left(e^{xy}\right)=??$

Solution:

$\frac{\partial ^3}{\partial \:y\partial \:x^2}\left(e^{xy}\right)=\frac{\partial ^2}{\partial \:x^2}[\frac{\partial }{\partial y}\left(e^{xy}\right)]\\ \Rightarrow \frac{\partial }{\partial y}\left(e^{xy}\right)=e^{xy}x\\ =\frac{\partial ^2}{\partial \:x^2}[e^{xy}x]\\$

Now

$\frac{\partial }{\partial \:x}[e^{xy}x] =\frac{\partial \:}{\partial \:x}\left(e^{xy}\right)x+\frac{\partial \:}{\partial \:x}\left(x\right)e^{xy}\\ =e^{xy}yx+1\cdot \:e^{xy}\\ =e^{xy}xy+e^{xy}$

д? -[ exy + ету 2.2 lе t] = а Әr a (ey xy) + От = y (yev*x+ ev*) + ey = e*9 ху° + 2e* у д

So

$\frac{\partial ^3}{\partial \:y\partial \:x^2}\left(e^{xy}\right)=e^{xy}xy^2+2e^{xy}y$

put (x,y)=(1,2)

$\frac{\partial ^3}{\partial \:y\partial \:x^2}\left(e^{xy}\right)_{(1,2)}=e^{1*2}1*2^2+2e^{1*2}2=8e^2=59.11$

Answer:

$\frac{\partial ^3}{\partial \:y\partial \:x^2}\left(e^{xy}\right)_{(1,2)}=8e^2$

If you have any doubts reply in here,please rate the answer.Thank you.