Find the four second-order partial derivatives for $f(x, y)=9 x^{7} y^{6}+4 x^{5} y^{4}$.

Step 1 ：The function given is \(f(x, y)=9 x^{7} y^{6}+4 x^{5} y^{4}\).

Step 2 ：First, find the first-order partial derivatives with respect to x and y. The derivative with respect to x is \(f_x = 63x^{6}y^{6} + 20x^{4}y^{4}\) and the derivative with respect to y is \(f_y = 54x^{7}y^{5} + 16x^{5}y^{3}\).

Step 3 ：Next, find the second-order partial derivatives. The second derivative with respect to x is \(f_{xx} = 378x^{5}y^{6} + 80x^{3}y^{4}\) and the second derivative with respect to y is \(f_{yy} = 270x^{7}y^{4} + 48x^{5}y^{2}\).

Step 4 ：Then, find the mixed partial derivatives. The derivative with respect to x then y is \(f_{xy} = 378x^{6}y^{5} + 80x^{4}y^{3}\) and the derivative with respect to y then x is \(f_{yx} = 378x^{6}y^{5} + 80x^{4}y^{3}\).

Step 5 ：Finally, the four second-order partial derivatives for \(f(x, y)=9 x^{7} y^{6}+4 x^{5} y^{4}\) are \(\boxed{f_{xx} = 378x^{5}y^{6} + 80x^{3}y^{4}, f_{yy} = 270x^{7}y^{4} + 48x^{5}y^{2}, f_{xy} = 378x^{6}y^{5} + 80x^{4}y^{3}, f_{yx} = 378x^{6}y^{5} + 80x^{4}y^{3}}\).