Difficulty with the multivariate chain rule

Question

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.

Answer 1

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 $x$: $$\frac{d}{dx}f(u,v)=\frac{\partial f}{\partial u}.\frac{du}{dx}+\frac{\partial f}{\partial v}.\frac{dv}{dx}=f_ug'(x)+f_vh'(x).$$

Answer 2

Yes it's correct with the advice just given in the previous answer,

Note that the result is the same you would obtain assuming f(x) as function of one variable.

EG

$$f(x)=\sin x \cdot x^2\implies f'(x)=\cos x\cdot x^2+\sin x\cdot 2x$$

$$f(\sin x, x^2)\implies f'(\sin x, x^2)=x^2\cdot\cos x+\sin x\cdot2x$$