Find the four second-order partial derivatives for $f(x, y)=8 x^{8} y^{7}+6 x^{5} y^{6}$
\[
f_{x x}=
\]
\[
f_{y y}=
\]
\[
f_{x y}=
\]
\[
f_{y x}=
\]

Step 1 ：Given the function \(f(x, y)=8 x^{8} y^{7}+6 x^{5} y^{6}\), we are asked to find the four second-order partial derivatives.

Step 2 ：The second-order partial derivatives are the derivatives of the first-order partial derivatives. The first-order partial derivatives are found by differentiating the function with respect to one variable while holding the other variable constant. The second-order partial derivatives are found by differentiating the first-order partial derivatives with respect to the other variable.

Step 3 ：The four second-order partial derivatives are: \(f_{xx}\), the second derivative of \(f\) with respect to \(x\); \(f_{yy}\), the second derivative of \(f\) with respect to \(y\); \(f_{xy}\), the derivative of \(f\) with respect to \(x\) and then \(y\); and \(f_{yx}\), the derivative of \(f\) with respect to \(y\) and then \(x\).

Step 4 ：By applying the rules of differentiation, we find that \(f_{xx}=8x^{3}y^{6}(56x^{3}y + 15)\), \(f_{yy}=12x^{5}y^{4}(28x^{3}y + 15)\), \(f_{xy}=4x^{4}y^{5}(112x^{3}y + 45)\), and \(f_{yx}=4x^{4}y^{5}(112x^{3}y + 45)\).

Step 5 ：\(\boxed{f_{xx}=8x^{3}y^{6}(56x^{3}y + 15)}\)

Step 6 ：\(\boxed{f_{yy}=12x^{5}y^{4}(28x^{3}y + 15)}\)

Step 7 ：\(\boxed{f_{xy}=4x^{4}y^{5}(112x^{3}y + 45)}\)

Step 8 ：\(\boxed{f_{yx}=4x^{4}y^{5}(112x^{3}y + 45)}\)