math.derivative returns a strange form of a polynomial involving negative powers.

[deSelbyPsyche]

Describe the bug
The bug is that when using math.derivative to evaluate 1x^2 -4x^-2, mathjs returns: (2 * x ^ 4 + 8) / x ^ 3. I appreciate that this is one valid way of notating the answer, however, it is quite strange. An analogy would be that if an arithmetic function evaluated 2+2 and returned 8/2 we would likely consider that a bug. It is true that 8/2 is one way of notating the answer but it's not how the vast majority of users would expect to receive the result, which would be 4.

It should be: 2 * x + 8 * x ^ -3, or 2 * x + 8 / x ^ 3. The documentation includes some examples of derivatives, but not expressions like this, so maybe it was overlooked.

As a comparison the Python library SymPy evaluates 1*x**2 -4*x**-2to: 2*x + 8/x**3.

To Reproduce
Call math.derivative with the above polynomial and see that it evaluates to the above result.

[josdejong]

Makes sense. I'll label this as a possible improvement since it is not strictly speaking a bug.

Anyone interested in looking into this?

[deSelbyPsyche]

Hey thanks for getting back to my comment. If you think it's a discussion and not a bug since it is a correct answer to the question that's cool.

If I get more time I may come back to this and try to implement a change, but for now I have moved to another approach and don't have time.

[gwhitney]

As a sort of workaround at the moment,
simplifyConstant(derivative("1x^2 -4x^-2", "x", {simplify: false})) returns 2 * x ^ 1 - -8 * x ^ (-3), which is almost, but not quite, what you want. The issue is that simplifyConstant literally only simplifies the constant subexpressions; it can't change subtraction of a negative into adding a positive. The simplify function can and does, but it assumes that you would prefer a rational function of polynomials with positive exponents: note simplify("2 * x ^ 1 - -8 * x ^ (-3)") produces (2 * x ^ 4 + 8) / x ^ 3. So there are subtle questions here, resulting from the fact that there's no absolute definition of when one expression is more "simple" than another. Is simplify "wrong" to change 2x + 8x⁻³ to (2x⁴+8)/x³? Should only rationalize do that, and simplify leave it alone? Or, if we like polynomial/monomial for the result of simplify, then is there a different set of simplification rules that we should apply in the case of simplifying a derivative in particular? (In the end, I agree that the output of derivative in this case looks "odd" and that 2x + 8x⁻³ would be more "normal".)