Does Maxima have problems differentiating integrals?

Question

When I ask Maxima for the value of

diff(integrate(f(y),y,0,x),x);

then it correctly derives that this expression is f(x). However, if I slightly modify the expression to

diff(integrate(f(y)^(1/2),y,0,x),x);

then Maxima asks be whether x is positive, zero, or negative. Answering positive or negative leads to the correct and same result of f(x)^(1/2). Answering zero gives an error because deriving by a constant is not well-defined.

Is this a limitation of Maxima or is there a way to get Maxima to get the right result without asking for the sign of x?

I have version 5.41.0 of Maxima and am using it via version 18.02.0 of wxMaxima.

score 1 · Accepted Answer · answered Jun 04 '21 at 02:56

Looks like the question is coming from integrate, not diff:

(%i2) integrate (f(y), y, 0, x);
                            x
                           /
                           [
(%o2)                      I  f(y) dy
                           ]
                           /
                            0
(%i3) integrate (sqrt(f(y)), y, 0, x);
Is x positive, negative or zero?

p;
                         x
                        /
                        [
(%o3)                   I  sqrt(f(y)) dy
                        ]
                        /
                         0
(%i4) integrate (sqrt(f(y)), y, 0, x);
Is x positive, negative or zero?

n;
                          0
                         /
                         [
(%o4)                  - I  sqrt(f(y)) dy
                         ]
                         /
                          x

Reordering the limits of integration is okay, although maybe not necessary, and it's inconsistent between %i2 and %i3. I guess that's a bug.

After that, diff has the expected effect:

(%i5) diff (%o2, x);
(%o5)                         f(x)
(%i6) diff (%o3, x);
(%o6)                      sqrt(f(x))
(%i7) diff (%o4, x);
(%o7)                      sqrt(f(x))

You can suppress the question by telling Maxima whether x is greater or less than zero. I don't know if that makes sense for the problem you are trying to solve.

(%i8) assume (x > 0);
(%o8)                        [x > 0]
(%i9) integrate (sqrt(f(y)), y, 0, x);
                         x
                        /
                        [
(%o9)                   I  sqrt(f(y)) dy
                        ]
                        /
                         0
(%i10) diff (%, x);
(%o10)                     sqrt(f(x))

Does Maxima have problems differentiating integrals?

1 Answers1