Verify that $f_{x y}=f_{y x}$ for the following function.
\[
f(x, y)=e^{x+y+16}
\]
\[
\begin{array}{l}
f_{x}=\square \\
f_{y}=\square \\
f_{x y}=f_{y x}=
\end{array}
\]

Answer

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Answer

Final Answer: $\boxed{f_{xy} = f_{yx} = e^{x+y+16}}$

Steps

Step 1 ：Given the function $f(x, y)=e^{x+y+16}$

Step 2 ：Find the first order partial derivatives $f_x$ and $f_y$

Step 3 ：$f_x = e^{x+y+16}$

Step 4 ：$f_y = e^{x+y+16}$

Step 5 ：Find the second order mixed partial derivatives $f_{xy}$ and $f_{yx}$

Step 6 ：$f_{xy} = e^{x+y+16}$

Step 7 ：$f_{yx} = e^{x+y+16}$

Step 8 ：Since $f_{xy} = f_{yx}$, we have verified the property for this function

Step 9 ：Final Answer: $\boxed{f_{xy} = f_{yx} = e^{x+y+16}}$