I am reading Elementary Calculus by Keisler and I am surprised as I arrive at the definition of partial derivatives (§ 11.3).
I was expecting a definition based on the standard part, similar to the definition given for the derivative of a function $f(x)$ of a single variable (§ 2.1). Very briefly (although actually deriving from the definition of the slope of $f(x)$ at $a$) : $$f'(x)=\mathrm{st}(\frac{f(x+\Delta x) - f(x)}{\Delta x}).$$
However, the definition of partial derivatives is introduced through the use of limits:
$$f_{x}(a,b)=\lim_{\Delta x\rightarrow 0}(\frac{f(a+\Delta x,b) - f(a,b)}{\Delta x}).$$
$$f_{y}(a,b)=\lim_{\Delta y\rightarrow 0}(\frac{f(a,b+\Delta y) - f(a,b)}{\Delta y}).$$
A few lines below, he identifies $f_{x}(a,b)$ and $f_y(a,b)$ with the standard parts ("when the derivatives exist") :
$$f_{x}(a,b)=\mathrm{st}(\frac{f(a+\Delta x,b) - f(a,b)}{\Delta x}),$$
$$f_{y}(a,b)=\mathrm{st}(\frac{f(a,b) - f(a,b+\Delta y)}{\Delta y}).$$
Instead of using limits, it seems the above two equations could have been used as definitions, provided the standard part to be the same for any infinitesimal $\Delta x$ or $\Delta y$ respectively.
But the fact that the definition is not directly based on the use of infinitesimals and standard parts (whereas a great deal have been made to properly define these concepts and ,up to then, avoid the use of limits) makes me think there might be a subtlety that is not explained and that goes over my head there. Is my feeling correct, and if so what is the subtlety that I am missing here?
Based on the definition of limit that Keisler gives, his definition of partial derivatives and yours are logically equivalent.
However, stating yours as written requires more text to explain that the standard part should be the same for all choices of infinitesimal $\Delta x$ (and $\Delta x\ne0$). In contrast, the limit notation is more compact, and reinforces real limits which are useful in a variety of contexts and used in other texts, etc.
I can only speculate about the reasons Keisler wrote things this way, but those are a couple reasons why I would have done the same.