What is difference between derivative in standard and non standard analysis?

Question

I am reading the book on complex analysis by Tristan Needham. In that book he explains derivative in an intuitive way as a quantity by which dx is expanded to get dy in both complex and real number plane. But In non standard analysis, derivative is defined in other way, Sh[(f(x+dx)-f(x)/dx)]=sh(dy/dx) where 'sh' denotes the shadow function Now is the definition of derivative given in the book of Tristan needham false, Because of the shadow function according to non standard analysis? In simple words my question is what is intuitive definition of derivative as given in book of Tristan needham according to non standard analysis?

Answer 1

The intuition for the derivative in Needham's Visual Complex Analysis is intuition, and so it is independent of the formal definition. Since the formal definition of the derivative in non-standard analysis is logically equivalent to the formal definition in standard analysis, Needham's intuition applies equally well.

Edit: Specifically, for $u$ that depends on $x$ but not on $y$, Needham (paraphrased) writes the following, using "infinitesimal" in an informal way.

…an infinitesimal…$dx$ the infinitesimal [displacement] $du$…$du=$ total change in $u$ due to moving along $dx$ $=$ (rate of change of $u$ with $x$)$\cdot$(change $dx$ in $x$) $=(\partial u/\partial x)dx$

This is reminiscent of the definition of the derivative in nonstandard analysis: If $dx$ is an infinitesimal (now used formally) change in $x$, and $u=f(x)$, then the corresponding change in $u$ is $du:={}^*f(x+dx)-{}^*f(x)$, and we have $\dfrac{\partial u}{\partial x}=\mathrm{sh}\left(\dfrac{du}{dx}\right)$ when the left side is defined.