Find the four second-order partial derivatives for $f(x,y)=8x^8{y}^7+6x^5{y}^6$
$$f_{xx}=$$
$$f_{yy}=$$
$$f_{xy}=$$
$$f_{yx}=$$

Step 1 ：Given the function $f(x,y)=8x^8{y}^7+6x^5{y}^6$, 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 ：$\boxed{{\displaystyle f_{xx}=8x^3{y}^6(56x^{3}y+15)}}$

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

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

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