Let $f: (0,1) \times (0,\infty) \times (0,\infty) \to \mathbb{R}$, $ f \in C^{1}$. I want to know if the following relationship is true by the chain rule. I don't know how to apply the chain rule, I think.
$$ f_{x}(x,s,t)f_{x,t}(x,s,t) + f_{x}(x,s,t)f_{x,s}(x,s,t) = \dfrac{d}{2dt}|f_{x}(x,s,t)|^{2} $$ I think that $$ \dfrac{d}{2dt}|f_{x}(x,s,t)|^{2} = f_{x}(x,s,t)f_{x,t}(x,s,t) $$
I'm confusing myself
The second equality is correct, assuming the necessary derivatives exist. The first equality could potentially hold if you plug in $s=t$ or with some funky $t$ dependence. When differentiating w.r.t. $t$ we can mostly assume that all other variables are constant (if you know they're not it should be explicitly stated).
PS: $\frac{d}{dt}$ isn't really a fraction, so you shouldn't write $2$ in the denominator.