ChatGPT Explains the Total Derivative

Patrick Reany

(4/16/2023)

Introduction: In physics literature the notion of a total derivative being the sum of the implicit
derivative and the explicit derivative is well known and accepted, largely because to a physicist,
these latter two derivatives have profound differences in meaning. However, in mathematical literature,
the distinction is not often described in those terms. So, I wanted to know where my two favorite
bots aligned on the subject. First, I asked BingChat to explain it, but it didn't even know what I was
talking about, because it interpreted it as a derivative distributing over a sum. However, ChatGPT
not only knew what I was talking about, it was able to demonstrate when asked to do so.

Patrick: Are you aware that a total derivative is equal to the sum of an explicit derivative and
an implicit derivative?

ChatGPT: Yes, I am aware of the concept of a total derivative and its relation to explicit
and implicit derivatives.

In calculus, the total derivative of a function of several variables is the sum of its partial
derivatives with respect to each variable, multiplied by the corresponding differentials.
This can also be expressed using the chain rule, where the total derivative is the sum of the
explicit derivative (the derivative of the function with respect to one of its variables) and the
implicit derivative (the derivative of the variable with respect to another variable, multiplied
by the corresponding partial derivative).

In other words, the total derivative takes into account all of the different ways in which a
function can change, both explicitly and implicitly.

ChatGPT: Sure, I'd be happy to demonstrate using the total derivative of the function
y = f(x,g(x)).

The total derivative of y with respect to x is given by:

dy/dx = ∂f/∂x + (∂f/∂g) * (dg/dx)

Here, the first term on the right-hand side is the explicit derivative of f with respect to x,
and the second term is the implicit derivative of f with respect to g, multiplied by the
derivative of g with respect to x.

To see this, let's take the partial derivative of y with respect to x:

∂y/∂x = ∂f/∂x + (∂f/∂g) * (∂g/∂x)

This partial derivative tells us how much y changes when we change x, while holding g
constant. But what if we allow g to change as well? To get the total derivative, we need
to take into account both of these ways that y can change.

If we think of g as a function of x, then the chain rule tells us that:

dy/dx = ∂f/∂x + (∂f/∂g) * (dg/dx)

This formula gives us the total derivative of y with respect to x, taking into account both
the explicit and implicit ways that y can change. The first term on the right-hand side tells
us how much y changes when we change x directly, while the second term tells us how
much y changes when we change x indirectly through its effect on g.

So, in summary, the total derivative of y = f(x,g(x)) with respect to x is the sum of the
explicit derivative of f with respect to x and the implicit derivative of f with respect to g,
multiplied by the derivative of g with respect to x.

Final Comments: This is very impressive of ChatGPT. I give its reply five points out of five.

I intentionally did not write down y(x) because I wanted ChatGPT to figure that out on its own,
which it did. So, how does a machine reason so as to turn an under-defined problem into a properly
defined problem? Well, I know how I do it. I follow the rule I made up long ago

Make any assumption necessary to solve the given problem in a reasonable amount of time,
using a reasonable amount of effort.

By the way, ChatGPT was not incorrect to refer to the total derivative either as ∂y/∂x or as dy/dx.
But if I had given it the 'function' y = y(x,g(x)), it would have to be more careful, just as humans
would have to be more careful in that case.