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

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Accepted Answer

# 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 $

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Question

Explanation:

Step 1: Find \( R_x \)

Step 2: Find \( R_{xx} \)

Step 3: Find \( R_{xy} \)

Step 4: Find \( R_y \)

Step 5: Find \( R_{yx} \)

Step 6: Find \( R_{yy} \)

Answer: