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Essay 10 points
Find the four second partial derivatives of

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01:09:46 Time Remaining 5 Essay 10 points Find the four second partial derivatives of $$ f(x, y)=y x^{4}+x y^{5}-9 $$ You may just state your answers. Edit View Insert Format Tools Table p Previous

UpStudy AI · Accepted Answer

The function is 
$$
f(x,y)=yx^4+xy^5-9.
$$

**Step 1. Compute the first partial derivatives.**

Differentiate with respect to $$ x $$:
$$
f_x(x,y)=\frac{\partial}{\partial x}(yx^4)+\frac{\partial}{\partial x}(xy^5)-\frac{\partial}{\partial x}(9)=4yx^3+y^5.
$$

Differentiate with respect to $$ y $$:
$$
f_y(x,y)=\frac{\partial}{\partial y}(yx^4)+\frac{\partial}{\partial y}(xy^5)-\frac{\partial}{\partial y}(9)=x^4+5xy^4.
$$

**Step 2. Compute the second partial derivatives.**

Differentiate $$ f_x(x,y)=4yx^3+y^5 $$ with respect to $$ x $$ to get $$ f_{xx} $$:
$$
f_{xx}(x,y)=\frac{\partial}{\partial x}(4yx^3+y^5)=4y\cdot 3x^2=12yx^2.
$$

Differentiate $$ f_x(x,y)=4yx^3+y^5 $$ with respect to $$ y $$ to get $$ f_{xy} $$:
$$
f_{xy}(x,y)=\frac{\partial}{\partial y}(4yx^3+y^5)=4x^3+5y^4.
$$

Differentiate $$ f_y(x,y)=x^4+5xy^4 $$ with respect to $$ y $$ to get $$ f_{yy} $$:
$$
f_{yy}(x,y)=\frac{\partial}{\partial y}(x^4+5xy^4)=0+5x\cdot 4y^3=20xy^3.
$$

Differentiate $$ f_y(x,y)=x^4+5xy^4 $$ with respect to $$ x $$ to get $$ f_{yx} $$:
$$
f_{yx}(x,y)=\frac{\partial}{\partial x}(x^4+5xy^4)=4x^3+5y^4.
$$

**Final Answers:**

- $$ f_{xx}(x,y)=12yx^2 $$
- $$ f_{xy}(x,y)=4x^3+5y^4 $$
- $$ f_{yy}(x,y)=20xy^3 $$
- $$ f_{yx}(x,y)=4x^3+5y^4 $$
The four second partial derivatives of $$ f(x, y) = yx^4 + xy^5 - 9 $$ are: - $$ f_{xx}(x,y) = 12yx^2 $$ - $$ f_{xy}(x,y) = 4x^3 + 5y^4 $$ - $$ f_{yy}(x,y) = 20xy^3 $$ - $$ f_{yx}(x,y) = 4x^3 + 5y^4 $$

01:09:46
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Essay 10 points
Find the four second partial derivatives of

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01:09:46 Time Remaining 5 Essay 10 points Find the four second partial derivatives of You may just state your answers. Edit View Insert Format Tools Table p Previous

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01:09:46
Time Remaining
5

Essay 10 points
Find the four second partial derivatives of

You may just state your answers.
Edit View Insert Format Tools Table
p
Previous