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How do you calculate the partial derivatives for a function of this sort using the chain rule? $$z=z(x,y)=f(3xy,\sin (xy))$$ where $f$ is a differentiable function from $\mathbb R^2$ to $\mathbb R$.

I want to use the form of the chain rule at the very bottom of this page, with partials. http://mathonline.wikidot.com/gradients-jacobian-matrices-and-the-chain-rule-review

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With $g(x,y)=3xy$ and $h(x,y)= \sin (xy)$ we have

$$z(x,y)=f(g(x,y),h(x,y)).$$

The chain rule gives

$$z_x(x,y)=f_x(g(x,y),h(x,y))g_x(x,y)+f_y(g(x,y),h(x,y))h_x(x,y)$$

and

$$z_y(x,y)=f_x(g(x,y),h(x,y))g_y(x,y)+f_y(g(x,y),h(x,y))h_y(x,y).$$

Now compute the partial derivatives $g_x, h_x, g_y$ and $h_y$ and proceed.

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