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

Question

Accepted 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. $80xy^{3}-105x^{2}y^{4}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

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

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

Answer: