find all second - order partial derivatives for the following.\n$r(x,y)=5x^{5}-2x^{6}y^{7}+8x^{5}y^{4}$\n$r_{…

Explanation:

Step 1: Find ( R_x )

Differentiate ( R(x,y) = 5x^5-2x^6y^7 + 8x^5y^4 ) with respect to ( x ): [ \begin{align*} R_x&=\frac{\partial R}{\partial x}\ &=5\times5x^4-2\times6x^5y^7+8\times5x^4y^4\ &=25x^4 - 12x^5y^7+40x^4y^4 \end{align*} ]

Step 2: Find ( R_{xx} )

Differentiate ( R_x = 25x^4 - 12x^5y^7+40x^4y^4 ) with respect to ( x ): [ \begin{align*} R_{xx}&=\frac{\partial^2 R}{\partial x^2}\ &=25\times4x^3-12\times5x^4y^7 + 40\times4x^3y^4\ &=100x^3-60x^4y^7 + 160x^3y^4 \end{align*} ]

Step 3: Find ( R_{xy} )

Differentiate ( R_x = 25x^4 - 12x^5y^7+40x^4y^4 ) with respect to ( y ): [ \begin{align*} R_{xy}&=\frac{\partial^2 R}{\partial y\partial x}\ &=-12x^5\times7y^6+40x^4\times4y^3\ &=-84x^5y^6 + 160x^4y^3 \end{align*} ]

Step 4: Find ( R_y )

Differentiate ( R(x,y) = 5x^5-2x^6y^7 + 8x^5y^4 ) with respect to ( y ): [ \begin{align*} R_y&=\frac{\partial R}{\partial y}\ &=-2x^6\times7y^6+8x^5\times4y^3\ &=-14x^6y^6+32x^5y^3 \end{align*} ]

Step 5: Find ( R_{yx} )

Differentiate ( R_y=-14x^6y^6 + 32x^5y^3 ) with respect to ( x ): [ \begin{align*} R_{yx}&=\frac{\partial^2 R}{\partial x\partial y}\ &=-14\times6x^5y^6+32\times5x^4y^3\ &=-84x^5y^6+160x^4y^3 \end{align*} ]

Step 6: Find ( R_{yy} )

Differentiate ( R_y=-14x^6y^6 + 32x^5y^3 ) with respect to ( y ): [ \begin{align*} R_{yy}&=\frac{\partial^2 R}{\partial y^2}\ &=-14x^6\times6y^5+32x^5\times3y^2\ &=-84x^6y^5 + 96x^5y^2 \end{align*} ]

Answer:

( R_{xx}=100x^3-60x^4y^7 + 160x^3y^4 ), ( R_{xy}=-84x^5y^6 + 160x^4y^3 ), ( R_{yx}=-84x^5y^6+160x^4y^3 ), ( R_{yy}=-84x^6y^5 + 96x^5y^2 )