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

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

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

Answer

Explanation:

Step1: Find the first - order partial derivative (f_x)

Use the power rule ((x^n)^\prime=nx^{n - 1}) and ((y^m)^\prime = 0) (when differentiating with respect to (x)). [ \begin{align*} f(x,y)&=10x^{2}y^{4}-7x^{3}y^{5}\ f_x&=\frac{\partial f}{\partial x}\ &=10\times2x^{2 - 1}y^{4}-7\times3x^{3 - 1}y^{5}\ &=20xy^{4}-21x^{2}y^{5} \end{align*} ]

Step2: Find the second - order partial derivative (f_{xy})

Differentiate (f_x) with respect to (y). Use the power rule ((y^n)^\prime=ny^{n - 1}) and ((x^m)^\prime = 0) (when differentiating with respect to (y)). [ \begin{align*} f_{xy}&=\frac{\partial f_x}{\partial y}\ &=20x\times4y^{4 - 1}-21x^{2}\times5y^{5 - 1}\ &=80xy^{3}-105x^{2}y^{4} \end{align*} ]

Answer:

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