How do I apply the chain rule to a double partial derivative of a multivariable function?
I know how to apply the chain rule to multivariable functions, however I need to differentiate twice with respect to a variable using the chain rule. Could somebody show me the way to do it? A general formula would suffice.
As an example I propose
$$\begin{align} f(x,y) &= e^{xy} \\ g(x,y) &= f\left(\sin\left(x^2 + y\right),x^3 + 2y + 1\right) \end{align}$$
Let’s compute $\displaystyle\frac{\partial^2 g}{\partial x^2}(0,0)$.
If we let $f$ be defined as $f(u,v)=e^{uv}$ instead, for clarity, then $$ \frac{\partial g}{\partial x}=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x} $$ Once you've calculated the first partial derivative, you repeat the above on said partial derivative to get the second derivative.