(1)

\begin{align*} \frac{df\left(g(x)\right)}{dx} & =\lim_{\Delta x\rightarrow0}\frac{f\left(g(x+\Delta x)\right)-f\left(g(x)\right)}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{f\left(g(x+\Delta x)\right)-f\left(g(x)\right)}{g(x+\Delta x)-g(x)}\frac{g(x+\Delta x)-g(x)}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{f\left(g(x)+\Delta g(x)\right)-f\left(g(x)\right)}{\Delta g(x)}\frac{g(x+\Delta x)-g(x)}{\Delta x}\cmt{\Delta g(x)=g(x+\Delta x)-g(x)}\\ & =f'(g(x))g'(x) \end{align*}

(2)

\begin{align*} \frac{df\left(g(x),h(x)\right)}{dx} & =\lim_{\Delta x\rightarrow0}\frac{f\left(g(x+\Delta x),h(x+\Delta x)\right)-f\left(g(x),h(x)\right)}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{f\left(g(x+\Delta x),h(x+\Delta x)\right)-f\left(g(x),h(x+\Delta x)\right)+f\left(g(x),h(x+\Delta x)\right)-f\left(g(x),h(x)\right)}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\left(\frac{f\left(g(x+\Delta x),h(x+\Delta x)\right)-f\left(g(x),h(x+\Delta x)\right)}{g(x+\Delta x)-g(x)}\frac{g(x+\Delta x)-g(x)}{\Delta x}+\frac{f\left(g(x),h(x+\Delta x)\right)-f\left(g(x),h(x)\right)}{h(x+\Delta x)-h(x)}\frac{h(x+\Delta x)-h(x)}{\Delta x}\right)\\ & =\lim_{\Delta x\rightarrow0}\left(\frac{f\left(g(x)+\Delta g,h(x+\Delta x)\right)-f\left(g(x),h(x+\Delta x)\right)}{\Delta g}\frac{g(x+\Delta x)-g(x)}{\Delta x}+\frac{f\left(g(x),h(x)+\Delta h\right)-f\left(g(x),h(x)\right)}{\Delta h}\frac{h(x+\Delta x)-h(x)}{\Delta x}\right)\\ & =\frac{\partial f\left(g(x),h(x)\right)}{\partial g(x)}g'(x)+\frac{\partial f\left(g(x),h(x)\right)}{\partial h(x)}h'(x) \end{align*}