On Jan 16, 2008 7:00 PM, David Joyner <[EMAIL PROTECTED]> wrote:
>
> Hi:
>
> I don't remember this topic coming up before but at some point
> it must be faced since it arises in Calculus 1. AFAIK, SAGE has no routines
> for
> (1) computing the "implicit derivative" y' if y is defined implicitly
> by F(x,y)=0,
> (2) plotting (x,y) subject to F(x,y)=0.
>
> (1): This is easy for SAGE to compute: y' = - F_x(x,y)/F_y(x,y).
> The problem is that it just isn't implemented yet to my knowledge.
> How should this be implemented? implicit_derivative(F(x,y),x) or something??
> Suggestions?
Is there a need for a special function? I would suggest to use regular
differentiation means.
For inspiration, I just tried this in sympy:
In [1]: g = Function("g")
In [2]: f(g(x), x)
Out[2]: f(g(x), x)
In [3]: e=f(g(x), x)
In [4]: e
Out[4]: f(g(x), x)
In [7]: e.args
Out[7]: (g(x), x)
In [8]: e.diff(x)
Out[8]:
d d d
─────(f(g(x), x))*──(g(x)) + ──(f(g(x), x))
dg(x) dx dx
In [9]: Basic.set_repr_level(1)
Out[9]: 2
In [10]: e.diff(x)
Out[10]: D(f(g(x), x), g(x))*D(g(x), x) + D(f(g(x), x), x)
(you need to use fixed width fonts to see [8], otherwise you'll just see a mess)
But I am not satisfied with the output, especially with
D(f(g(x), x), x)
since it is not clear if it means the total derivative, or just with
respect to the second parameter. Or do you mean something like this:
In [4]: e=f(x)**2+f(x)*x**3-3*f(x)
In [5]: e.diff(x)
Out[5]:
d 3 d d 2
- 3*──(f(x)) + x *──(f(x)) + 2*f(x)*──(f(x)) + 3*x *f(x)
dx dx dx
In [6]: Basic.set_repr_level(1)
Out[6]: 2
In [7]: e.diff(x)
Out[7]: -3*D(f(x), x) + x**3*D(f(x), x) + 2*f(x)*D(f(x), x) + 3*x**2*f(x)
In [9]: solve(e.diff(x).subs(f(x).diff(x), y) == 0, [y])
Out[9]: [3*x**2/(3 - x**3 - 2*f(x))*f(x)]
in [7] I differentiate the implicit definition of f(x), in [9] I solve
for the D(f(x), x) by substituting it for "y" and solving for y.
I cannot now figure out how to do this in Sage, but I am not sure what
the best interface to all of this is. But I think the less functions
(idioms) one needs to remember to do any calculus stuff in Sage, the
better.
Ondrej
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