Should $d/dx$ ever be used when solving for a partial derivative? Or would all instances of it be replaced with $\partial/\partial x$?

Question

You hold one of the variables constant and find the derivative of the other when trying to find the partial derivative, so wouldn't something like this be correct?:

$\frac{\partial}{\partial x}(|xy|) = \frac{\partial}{\partial x}(|x||y|) = |y| \frac{d}{dx}|x|=\frac{|y|x}{|x|}$

Or would this be correct:

$\frac{\partial}{\partial x}(|xy|) = \frac{\partial}{\partial x}(|x||y|) = |y| \frac{\partial}{\partial x}|x|=\frac{|y|x}{|x|}$

Or are either of those fine?

Any help is appreciated.

Answer 1

The basic problem: the Leibniz notation is confusing and deceptive. Wonderful quote from How misleading is it to regard $dy/dx$ as a fraction?:

Treating $dy/dx$ as a fraction is the gateway drug to treating $\partial y/\partial x$ as a fraction.

In your case, we can write (for $x\ne 0$!) $$\frac{d}{dx}|x| = \frac{\partial}{\partial x}|x|$$

without confusion because there are no more variables involved.