find all second - order partial derivatives for the following,\n\n r(x,y)=3x^{5}-8x^{7}y^{4}+5x^{6}y^{5}…

find all second - order partial derivatives for the following,\n\n r(x,y)=3x^{5}-8x^{7}y^{4}+5x^{6}y^{5} \n\n r_{xx}=\n\n r_{yy}=\n\n r_{xy}=\n\n r_{yx}=

find all second - order partial derivatives for the following,\n\n r(x,y)=3x^{5}-8x^{7}y^{4}+5x^{6}y^{5} \n\n r_{xx}=\n\n r_{yy}=\n\n r_{xy}=\n\n r_{yx}=

Answer

Explanation:

Step1: Find (R_x)

Differentiate (R(x,y)) with respect to (x): (R_x=\frac{\partial R}{\partial x}=3\times5x^{4}-8\times7x^{6}y^{4}+5\times6x^{5}y^{5}=15x^{4}-56x^{6}y^{4}+30x^{5}y^{5})

Step2: Find (R_{xx})

Differentiate (R_x) with respect to (x): (R_{xx}=\frac{\partial R_x}{\partial x}=15\times4x^{3}-56\times6x^{5}y^{4}+30\times5x^{4}y^{5}=60x^{3}-336x^{5}y^{4}+150x^{4}y^{5})

Step3: Find (R_y)

Differentiate (R(x,y)) with respect to (y): (R_y=\frac{\partial R}{\partial y}=-8\times4x^{7}y^{3}+5\times5x^{6}y^{4}=-32x^{7}y^{3}+25x^{6}y^{4})

Step4: Find (R_{yy})

Differentiate (R_y) with respect to (y): (R_{yy}=\frac{\partial R_y}{\partial y}=-32\times3x^{7}y^{2}+25\times4x^{6}y^{3}=-96x^{7}y^{2}+100x^{6}y^{3})

Step5: Find (R_{xy})

Differentiate (R_x) with respect to (y): (R_{xy}=\frac{\partial R_x}{\partial y}=-56\times4x^{6}y^{3}+30\times5x^{5}y^{4}=-224x^{6}y^{3}+150x^{5}y^{4})

Step6: Find (R_{yx})

Differentiate (R_y) with respect to (x): (R_{yx}=\frac{\partial R_y}{\partial x}=-32\times7x^{6}y^{3}+25\times6x^{5}y^{4}=-224x^{6}y^{3}+150x^{5}y^{4})

Answer:

(R_{xx}=60x^{3}-336x^{5}y^{4}+150x^{4}y^{5}) (R_{yy}=-96x^{7}y^{2}+100x^{6}y^{3}) (R_{xy}=-224x^{6}y^{3}+150x^{5}y^{4}) (R_{yx}=-224x^{6}y^{3}+150x^{5}y^{4})