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• Notes

Find the first partial derivatives of the function.
$f(x, y)=x^{2}+5 x y^{3}$

$f(x, y)=x^{2}+5 x y^{3}$

$f_{x}(x, y)=\frac{\partial}{\partial x}[f(x, y)]=2 x+5 y^{3}$

$f_{y}\left(x, y\right)=\frac{2}{2 y}\left[f\left(x, y\right)\right]=5 \times 3 x y^{2}=15 x y^{2}$

Find the first partial derivatives of the function.
$f(x, y, z)=x^{3} y z^{2}+2 y z$

$f(x, y, z)=x^{3} y z^{2}+2 y z$

$f_{x}\left(x, y, z\right)=\frac{\partial}{\partial x} f\left(x, y, z\right)=yz^{2}\left(3 x^{2}\right)=3 x^{2} y z^{2}$

$f_{y}(x, y, z)=\frac{\partial}{\partial y} f(x, y, z)=x^{3} z^{2}+2 z$

$f_{z}(x, y, z)=\frac{\partial}{\partial z} f(x, y, z)=x^{3} y(2 z)+2 y$
$=2 x^{3} y z + 2 y$
Find all the second partial derivatives.
$f(x, y)=x^{4} y-2 x^{3} y^{2}$

$f(x, y)=x^{4} y-2 x^{3} y^{2}$

$f_{x}\left({x, y}\right)=4 x^{3} y-6 x^{2} y^{2}$

$f_{y}\left(x, y\right)=x^{4}-4 x^{3} y$

$f_{x x}\left(x, y\right)=\frac{\partial}{\partial x} f_{x}\left(x, y\right)=12 x^{2} y-12 x y^{2}$

$f_{y y} (x, y)=\frac{\partial}{\partial y} f_{y} (x, y)=-4 x^{3}$

$f_{x y}(x, y)=\frac{\partial}{\partial y} f_{x}(x, y)=$

$=4 x^{3}-6 x^{2}[2 y]=4 x^{3}-12x^{2} y$

$f_{x, y} \frac{\partial}{\partial x}=\frac{\partial}{\partial x} f(x, y)$

$=\frac{\partial}{\partial x}\left[x^{4}-4 x^{3} y\right]$

$f_{y x} \left(x, y\right)=4 x^{3}-12 x^{2} y$