[Rafael-SOWNet]

Part of #233.

Every rule differentiated its parts by calling InnerDifferentiate directly, and only the base rule knew that a node without the variable differentiates to 0. So a power, a function, a logarithm or a quotient with no x in it went through the chain rule, and InnerSimplified folded 0 * f' into 0 provided <where f' is defined>. The derivative was NaN where the expression has a value:

.Differentiate("x") of	at	master and 2.5.0	this
(a - a)^(1/2)	—	0 provided not a - a = 0, NaN	0
sqrt(a) * x	a = 0	NaN	sqrt(a)
arcsin(a) * x	a = 1	NaN	arcsin(a)
ln(c - c), 1/(a - a)	—	NaN	0

The new DerivativeOver(Variable) answers 0 for a node without the variable. Nodes containing a set, a matrix, a provided, a piecewise or a written NaN are excluded and keep their own rules: {1, 2} still gives {0}, and a piecewise keeps its conditions. All 47 child calls now go through it. The protected InnerDifferentiate overrides keep their names, so the public API is unchanged. Docs/Contributing/AddingNode.cs now tells a new node to call DerivativeOver for its parts. Binder variables (#984, #993) are unaffected.

The integrator relied on the spurious conditions in four places, which are now explicit:

• SolveByEulerSubstitution's perfect-square test asked a value and missed symbolic squares such as (2 a b)^2 - 4 b^2 a^2. It now asks IsAPerfectSquareDiscriminant.
• A symbolic root in x(t) under the same substitution was in effect never taken, because dx/dt carried provided not a = 0. It is now declined explicitly; taken, acos(a + b x)/x^4 ran past the budget.
• Three routes the conditions cut short now decline early by value instead:
  • SolveByPartialFractions' logarithmic-derivative test (MayBeAConstantMultiple);
  • a candidate even in x in SolveBySubstitution, which repeats at −x;
  • a substitution whose quotient keeps x through every rewriting (KeepsTheVariableThroughEveryRewriting).
    Without them, 6.7.1:897 went from 4.1 GB to 20.8 GB; with them it is 0.38 GB.

Answers that change:

• 1/(x sqrt(k^2 x^2 + 2 k x + 1)): no value anywhere on master, declined by 2.5.0; now answered by the perfect-square rule, 832 characters.
• 1/(x (a^2 + 2 a b x^3 + b^2 x^6)^(5/2)) (Rubi's 1.2.3.2:131): answered on master in 1,039 characters with no value at any sample point; now declined, as 2.5.0 declined it.

Chain against master a567c8b, with the harness at its current revision on both arms:

• pocket: every 15th row of every family, every 8th of 1.2.x, every 3rd of 6.7.1 and 1.1.2.4, 4,231 rows in all;
• allocation over the 4,194 rows solved on both: 825.5 → 746.3 GB;
• 1.1.2.4:566 and 4.3.2.1:690 go from timeout to 1.6 s and 0.65 s, alone;
• 6.3.2:218 is answered by Integrate itself where master needed Simplify afterwards;
• samples of families 0–8 unchanged, and no row's allocation grew past 1.3×.

row, alone	master	this
6.7.1:897	4.13 GB	0.38 GB
6.7.1:891	6.68 GB	0.31 GB
1.2.2.5:70	2.277 GB	2.262 GB
1.1.2.4:548	1.555 GB	0.183 GB

All four answers are identical to master's.

Tests:

• 31 new cases in DerivativeTest.
• The three InnerSimplifyTest.PiecewiseDerivative2* tests now expect piecewise(0 provided True), which simplifies to 0. Their old not x^2 = 0 was the quotient rule's denominator for a 1/x that does not vary in y, and their own comment said it "does not collapse all the way to 0".
• A local run of the Calculus, derivative, limit, series and piecewise tests (6,010) passed except three at load 15:
  • InverseTangentPowerByPartsTest.ANonElementaryRemainderIsDeclinedQuickly;
  • two SlowDeclinesAheadOfTheAnsweringRuleIntegralTest cases.
    All three pass alone. Those two rows allocate the same on master (7.65 and 3.93 GB) and this branch (7.65 and 3.90 GB), with identical answers.

The BREAKING-CHANGES.md entry has 2.5.0's column measured. It also rewords one sentence of the acsch(c x) entry, which said a node with a rule of its own keeps it, which is no longer true.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura