# How do I apply the chain rule to double partial derivative of a multivariable function?

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 $\displaystyle\frac{\partial^2 g}{\partial x^2}(0,0)$.

• what do you mean by $sen(x^2+y)$ – Juniven May 10 '17 at 1:13

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.