I am reading a thermodynamics book. I am confused on how they say use the chain rule Here. it makes no sense to me how they go from dV/dt to (dV/ds)(ds/dt) . I know how the chain rule works ,just dont know where they got these values
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Well they use the chain rule in the normal way here. If V is a function of s and s is a function of t, V(s(t)), then the chain rule tells you that [itex] (V \circ s)'(t)=V'(s(t))s'(t)[/itex]. Is that more familiar? This is Identical to [tex]\frac{dV}{ds}\frac{ds}{dt}[/tex]. To display this a bit more clearly lets start with the function V(s(t)). We then use the chain rule by substituting u=s(t). So [tex](V \circ s)'(t)=V'(u)u'=\frac{d V}{d u} \frac{d u}{d t}=\frac{d V}{d s} \frac{d s}{d t}[/tex].