# How to apply the chain rule to a double partial derivative of a multivariable function? f(x,y)=e^(xy) g(x,y)=f (sin(x^2+y),x^2+2y+1 Let’s compute d^2g/dx^2(0,0)

How to apply the chain rule to a double partial derivative of a multivariable function?
$\begin{array}{rl}f\left(x,y\right)& ={e}^{xy}\\ g\left(x,y\right)& =f\left(\mathrm{sin}\left({x}^{2}+y\right),{x}^{3}+2y+1\right)\end{array}$
Let’s compute $\frac{{\mathrm{\partial }}^{2}g}{\mathrm{\partial }{x}^{2}}\left(0,0\right)$
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If we let $f$ be defined as $f\left(u,v\right)={e}^{uv}$ instead, for clarity, then
$\frac{\mathrm{\partial }g}{\mathrm{\partial }x}=\frac{\mathrm{\partial }f}{\mathrm{\partial }u}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }f}{\mathrm{\partial }v}\frac{\mathrm{\partial }v}{\mathrm{\partial }x}$
Once you've calculated the first partial derivative, you repeat the above on said partial derivative to get the second derivative.