find the second - order partial derivative find ( f_{xy} ) when ( f(x,y)=10x^{2}y^{4}-7x^{3}y^{5} ).\na. (…

find the second - order partial derivative find ( f_{xy} ) when ( f(x,y)=10x^{2}y^{4}-7x^{3}y^{5} ).\na. ( 80xy^{3}-21x^{2}y^{4} ) b. ( 160xy^{3}-105x^{2}y^{4} ) c. ( 80xy^{3}-105x^{2}y^{4} ) d. ( 160xy^{3}-21x^{2}y^{4} )

find the second - order partial derivative find ( f_{xy} ) when ( f(x,y)=10x^{2}y^{4}-7x^{3}y^{5} ).\na. ( 80xy^{3}-21x^{2}y^{4} ) b. ( 160xy^{3}-105x^{2}y^{4} ) c. ( 80xy^{3}-105x^{2}y^{4} ) d. ( 160xy^{3}-21x^{2}y^{4} )

Answer

Explanation:

Step1: Differentiate ( f(x,y) ) with respect to ( x )

Use the power rule (\frac{\partial}{\partial x}(x^n)=nx^{n - 1}) and (\frac{\partial}{\partial x}(y^m)=0) (treating ( y ) as a constant). [ \begin{align*} f_x&=\frac{\partial}{\partial x}(10x^{2}y^{4}-7x^{3}y^{5})\ &=10\times2x^{2 - 1}y^{4}-7\times3x^{3 - 1}y^{5}\ &=20xy^{4}-21x^{2}y^{5} \end{align*} ]

Step2: Differentiate ( f_x ) with respect to ( y )

Use the power rule (\frac{\partial}{\partial y}(y^n)=ny^{n - 1}) and (\frac{\partial}{\partial y}(x^m)=0) (treating ( x ) as a constant). [ \begin{align*} f_{xy}&=\frac{\partial}{\partial y}(20xy^{4}-21x^{2}y^{5})\ &=20x\times4y^{4 - 1}-21x^{2}\times5y^{5 - 1}\ &=80xy^{3}-105x^{2}y^{4} \end{align*} ]

Answer:

C. (80xy^{3}-105x^{2}y^{4})