Find the four second-order partial derivatives.
\[
f(x, y)=5 x^{6} y-2 x y+6 y
\]
\[
f_{x x}(x, y)=
\]
\[
f_{x y}(x, y)=
\]
\[
f_{y x}(x, y)=
\]
\[
f_{y y}(x, y)=
\]

Step 1 ：The given function is \(f(x, y)=5 x^{6} y-2 x y+6 y\).

Step 2 ：First, find the first-order partial derivatives \(f_x(x, y)\) and \(f_y(x, y)\).

Step 3 ：The first-order partial derivative with respect to \(x\) is \(f_x(x, y) = 30x^{5}y - 2y\).

Step 4 ：The first-order partial derivative with respect to \(y\) is \(f_y(x, y) = 5x^{6} - 2x + 6\).

Step 5 ：Next, find the second-order partial derivatives \(f_{xx}(x, y)\), \(f_{xy}(x, y)\), \(f_{yx}(x, y)\), and \(f_{yy}(x, y)\).

Step 6 ：The second-order partial derivative with respect to \(x\) twice is \(f_{xx}(x, y) = 150x^{4}y\).

Step 7 ：The second-order partial derivative with respect to \(x\) then \(y\) is \(f_{xy}(x, y) = 30x^{5} - 2\).

Step 8 ：The second-order partial derivative with respect to \(y\) then \(x\) is \(f_{yx}(x, y) = 30x^{5} - 2\).

Step 9 ：The second-order partial derivative with respect to \(y\) twice is \(f_{yy}(x, y) = 0\).

Step 10 ：\(\boxed{f_{xx}(x, y) = 150x^{4}y, f_{xy}(x, y) = 30x^{5} - 2, f_{yx}(x, y) = 30x^{5} - 2, f_{yy}(x, y) = 0}\)