Difficulty with the multivariate chain rule I am having a minor issue with the chain rule. I remember its properties but I just cannot quite remember if I am doing it correctly. If someone could just check if this identity is correct that would be great. It is the last thing I appealed to in a proof Ive been working on for leisure and I want to make sure I don't embarrass myself by rememvering wrong.
$(f(g(x),h(x)))' = f'(g(x),h(x)) * g'(x) + f'(g(x),h(x)) * h'(x)$
I know that each derivative of $f$ in above is the partial derivative with respect to each argument, but is that right? I just want to make sure I am not forgetting something vital.

Your calculation is almost okey, but instead of $f'(g(x), h(x))$ you have to write partial derivatives of each component. More precisely, it is better to let $u=g(x)$ and $v=h(x)$ and differentiate w.r. to $x$:  $$\frac{d}{dx}f(u,v)=\frac{\partial f}{\partial u}.\frac{du}{dx}+\frac{\partial f}{\partial v}.\frac{dv}{dx}=f_ug'(x)+f_vh'(x).$$
