22. Find the four second partial derivatives of the following function
\[
f(x, y)=y^{5} \sin 2 x
\]
\[
\begin{array}{l}
f_{x x}=\square \\
f_{x y}=\square \\
f_{y y}=\square \\
f_{y x}=\square
\end{array}
\]

Step 1 ：Given the function \(f(x, y)=y^{5} \sin 2x\)

Step 2 ：Find the first partial derivatives, \(f_x\) and \(f_y\), by differentiating \(f(x, y)\) with respect to x and y respectively, treating the other variable as a constant

Step 3 ：\(f_x = 2y^{5}\cos 2x\)

Step 4 ：\(f_y = 5y^{4}\sin 2x\)

Step 5 ：Find the second partial derivatives, \(f_{xx}\), \(f_{yy}\), \(f_{xy}\), and \(f_{yx}\), by differentiating \(f_x\) and \(f_y\) with respect to x and y respectively

Step 6 ：\(f_{xx} = -4y^{5}\sin 2x\)

Step 7 ：\(f_{xy} = 10y^{4}\cos 2x\)

Step 8 ：\(f_{yy} = 20y^{3}\sin 2x\)

Step 9 ：\(f_{yx} = 10y^{4}\cos 2x\)

Step 10 ：Final Answer: \(\boxed{f_{xx} = -4y^{5}\sin 2x, f_{xy} = 10y^{4}\cos 2x, f_{yy} = 20y^{3}\sin 2x, f_{yx} = 10y^{4}\cos 2x}\)