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$f\left(x\,y\right)\=\sqrt{3 x^2\+y^3}$$\stackrel{\text{assign as function}}{\to }$$f$

Obtain the second-degree Taylor polynomial

Obtain the second-degree Taylor polynomial from first principles

$\mathrm{f\_\_x}\left(x\,y\right)\=\frac{\partial }{\partial x} f\left(x\,y\right)$$\stackrel{\text{assign as function}}{\to }$$\mathrm{f\_\_x}$

$\mathrm{f\_\_y}\left(x\,y\right)\=\frac{\partial }{\partial y} f\left(x\,y\right)$$\stackrel{\text{assign as function}}{\to }$$\mathrm{f\_\_y}$

$$\begin{gathered} f\left(4\,1\right)\+\left(\mathrm{f\_\_x}\left(4\,1\right)\cdot \left(x-4\right)\+\mathrm{f\_\_y}\left(4\,1\right)\cdot \left(y-1\right)\right) \\ \+\frac{1}{2}\left(\mathrm{f\_\_xx}\left(4\,1\right)\cdot {\left(x-4\right)}^2\+2 \mathrm{f\_\_xy}\left(4\,1\right)\cdot \left(x-4\right)\cdot \left(y-1\right)\+\mathrm{f\_\_yy}\left(4\,1\right)\cdot {\left(y-1\right)}^2\right) \end{gathered}$$

$\sqrt{49}\+\frac{12}{49}\sqrt{49}\left(x-4\right)\+\frac{3}{98}\sqrt{49}\left(y-1\right)\+\frac{3}{4802}\sqrt{49}{\left(x-4\right)}^2-\frac{18}{2401}\sqrt{49}\left(x-4\right)\left(y-1\right)\+\frac{579}{19208}\sqrt{49}{\left(y-1\right)}^2$

$\stackrel{\text{simplify}}{\=}$

$\frac{594}{343}x\+\frac{3}{1372}y-\frac{1}{2744}\+\frac{3}{686}{x}^2-\frac{18}{343}yx\+\frac{579}{2744}{y}^2$

$f≔\left(x\,y\right)\to \sqrt{3 x^2\+y^3}\:$

$p≔\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{TaylorApproximation}\left(f\left(x\,y\right)\,\left[x\,y\right]\=\left[4\,1\right]\,2\right)$

$\frac{12}{7}x\+\frac{3}{14}y-\frac{1}{14}\+\frac{3}{686}{\left(x-4\right)}^2-\frac{18}{343}\left(y-1\right)\left(x-4\right)\+\frac{579}{2744}{\left(y-1\right)}^2$

$\mathrm{simplify}\left(p\right)$

$\frac{594}{343}x\+\frac{3}{1372}y-\frac{1}{2744}\+\frac{3}{686}{x}^2-\frac{18}{343}yx\+\frac{579}{2744}{y}^2$

Obtain the second-degree Taylor polynomial from first principles

$$\begin{gathered} q≔f\left(4\,1\right)\+\left({\mathrm{D}}_1\left(f\right)\left(4\,1\right)\cdot \left(x-4\right)\+{\mathrm{D}}_2\left(f\right)\left(4\,1\right)\cdot \left(y-1\right)\right) \\ \+\frac{1}{2}\left({\mathrm{D}}_{1\,1}\left(f\right)\left(4\,1\right)\cdot {\left(x-4\right)}^2\+2 {\mathrm{D}}_{1\,2}\left(f\right)\left(4\,1\right)\cdot \left(x-4\right)\cdot \left(y-1\right)\+{\mathrm{D}}_{2\,2}\left(f\right)\left(4\,1\right)\cdot {\left(y-1\right)}^2\right) \end{gathered}$$

$7\+\frac{12}{49}\sqrt{49}\left(x-4\right)\+\frac{3}{98}\sqrt{49}\left(y-1\right)\+\frac{3}{4802}\sqrt{49}{\left(x-4\right)}^2-\frac{18}{2401}\sqrt{49}\left(x-4\right)\left(y-1\right)\+\frac{579}{19208}\sqrt{49}{\left(y-1\right)}^2$

$\mathrm{simplify}\left(q\right)$

$\frac{594}{343}x\+\frac{3}{1372}y-\frac{1}{2744}\+\frac{3}{686}{x}^2-\frac{18}{343}yx\+\frac{579}{2744}{y}^2$