Can ∆ always be replaced with d? Can one always replace the "big change" (∆) with "small change"(d)?
For instance,
∆G=∆H-T∆S to
dG=dH- TdS. 
Then can I write ∆ as d when the thing in front of it is a differentiable function So that it's infinitesimal change makes sense?
 A: The example you posted with "Δ" is Gibbs' Free Energy. However, when we put "d" in it, we get the first law for reversible processes. These are not the same equation, and it's a coincidence they're so similar.
In general, Δ and d mean different things and aren't usually interchangeable.
A: To put it simply, you replace it with $d$, as the change goes to zero.
But you should know that this is a pretty technical object in math, and sometimes there are disagreements by math instructors regarding what $d$ really is, i.e., there are some ways to define what $dx$ or $dt$ is that some mathematicians may not agree with.
A: $\Delta x$ is usually a real number, while the $dx$ in differentiation $\frac{dy}{dx}$  and integration $\int f\,dx$ are not. $dx$ is simply a shorthand for "variable that is integrating with respect to".
The symbol $dx$ follows intuitively from the definition of Riemann sum, (integrating $f$ on $[a,b]$)
$$ \lim_{n \to \infty} \sum_{k=0}^{n-1} f(x_k^*) \Delta x = \int_a^b f\,dx
$$
but it is only a symbol.
A: Different notations are used for conventions.You may use anything you want as long as you specify the correct meaning of what you are using.
But,try avoiding use of $d$ as $\displaystyle\lim_{x\rightarrow0}∆x=dx$.
